In the video, Mr Trump says "you can do anything" to women "when you're a star" and brags about trying to grope and kiss women. He has since apologised.
Mr Trump has clearly been an unpredictable candidate but no previous revelation or off-the-cuff comment has generated this much reaction.
So who in the Republican camp is still backing him and who has decided enough is enough?
The Utah governor, who previously supported Mr Trump, tweeted that the comments he had made were "beyond offensive and despicable". He continued: "While I cannot vote for Hillary Clinton, I will not vote for Trump."
A congressman for Utah, he withdrew support live on television.
He said: "It's sad really, but I can't endorse Donald Trump for president after those comments and the way he said them.
"I can't look my 15-year-old daughter in the eye and tell her I endorse this man to become president."
Like his Utah colleague Mr Herbert, he said the video left him at a loss regarding who to vote for, as there was "no way" he would vote for Hillary Clinton.
Mr Chaffetz is the chairman of the committee which is investigating Mrs Clinton's use of emails while she was secretary of state.
She was the only woman who ran for the Republican candidacy for 2016 and was briefly the vice-presidential candidate on Ted Cruz's ticket.
Although she once said she was "horrified" by Mr Trump, in September she finally gave him her support - but only as an alternative to Mrs Clinton.
In response to the tape, she said: "We must have a conservative in the White House to restore accountability, opportunity and security. For the sake of our Constitution and the rule of law, we must defeat Hillary Clinton. Today I ask Donald Trump to step aside and for the RNC to replace him with Gov Mike Pence."
The prominent senator for Arizona and one-time presidential nominee said Mr Trump should "suffer the consequences" of his remarks.
"Donald Trump's behaviour make(s) it impossible to continue to offer even conditional support for his candidacy," he said.
"There are no excuses for Donald Trump's offensive and demeaning comments.
"No woman should ever be victimised by this kind of inappropriate behaviour. He alone bears the burden of his conduct and alone should suffer the consequences."
The former secretary of state wrote on her Facebook page: "Enough! Donald Trump should not be President. He should withdraw.
"As a Republican, I hope to support someone who has the dignity and stature to run for the highest office in the greatest democracy on Earth."
The former California governor, who is more famous for his acting career and who took over from Mr Trump as a host of The Apprentice on NBC, said he would not vote for his predecessor in the presidential election.
He said: "I want to take a moment to remind my fellow Republicans that it is not only acceptable to choose your country over your party - it is your duty."
The former governor of New York tweeted that he was "horrified" at the tape and that Mr Trump's campaign was "a poisonous mix of bigotry and ignorance".
"Enough!" Mr Pataki tweeted. "He needs to step down."
A former governor of Minnesota, he said the comments were "unacceptable, and disqualifying for someone who hopes to serve as Commander in Chief".
A former Utah governor, Mr Huntsman called for the Republican vice-presidential candidate to take over running for president instead.
"In a campaign cycle that has been nothing but a race to the bottom, at such a critical moment for our nation, the time has come for Governor Pence to lead the ticket," he told the Salt Lake City Tribune in his home state.
Another Utah Republican, the senator addressed Mr Trump with the words: "With all due respect sir, you, sir, are the distraction. Your conduct, sir, is the distraction. It's a distraction from the very principles that will help us win in November."
He continued: "I respectfully ask you, with all due respect, to step aside, step down, allow someone else to carry the banner of these principles, these principles that have made our country great, these principles that will stand as a beacon of hope to the American family."
The Colorado congressman was one of the first Republicans to call on Mr Trump to stop running for president after the tapes of his comments were broadcast.
On Friday, he said: "For the good of the country, and to give the Republicans a chance of defeating Hillary Clinton, Mr. Trump should step aside.
"His defeat at this point seems almost certain."
An Alabama senator, she said: "As disappointed as I've been with his antics throughout this campaign, I thought supporting the nominee was the best thing for our country and our party.
"Now, it is abundantly clear that the best thing for our country and our party is for Trump to step aside and allow a responsible, respectable Republican to lead the ticket."
A New Jersey congressman, he said: "I cannot support and will not vote for Donald Trump or Hillary Clinton to be president of the United States. I will write in Governor Mike Pence for President."
The Virginia congresswoman, who had not yet declared support for Mr Trump, said she could not "in good conscience" vote for him.
She echoed calls for somebody else to replace him, saying: "This is disgusting, vile, and disqualifying.
"No woman should ever be subjected to this type of obscene behaviour and it is unbecoming of anybody seeking high office.
"In light of these comments, Donald Trump should step aside and allow our party to replace him with Mike Pence or another appropriate nominee from the Republican Party."
A senator for Alaska who had not yet declared her support for Mr Trump either, she said she watched the video immediately after a meeting about sexual exploitation and she felt "disgusted".
She said Mr Trump had "forfeited the right" to be the Republican party's nominee.
A Nebraska senator, she said: "The comments made by Mr Trump were disgusting and totally unacceptable under any circumstance. It would be wise for him to step aside and allow Mike Pence to serve as our party's nominee."
The Idaho senator also called for another candidate to be put forward instead, and said: "I have reached the decision that I can no longer endorse Donald Trump. This is not a decision that I have reached lightly, but his pattern of behaviour has left me no choice."
A Utah congressman, he said: "I'm incredibly disappointed in our party's candidate. And unlike the Democrats who have proven completely unwilling to hold secretary Clinton accountable for her illegal activities that endangered our national security, I am willing to hold Mr Trump accountable.
"I am therefore calling for him to step aside and to allow Mike Pence to lead our party."
An Arizona senator, he tweeted that "America deserves far better" than Mr Trump and that he should withdraw from the race.
A Nebraska congressman, he called for Mr Trump to step aside "and allow Mike Pence to become the Republican nominee".
A New Jersey congressman, he said: "Donald Trump's comments are inexcusable. I am appalled that he would brag about violating a woman's physical boundaries.
"As a husband and father of two daughters, I denounce his comments and the behaviour that it incites. I believe that Mike Pence would be the best nominee for the Republican Party to defeat Hillary Clinton."
A Nevada congressman, he had previously said he was "100%" behind Mr Trump.
But on Saturday, he pulled his support, saying: "I think that when we degrade that mother, wife, housewife, whatever you want to deal with, daughter - that you degrade America."
Another Nevada congressman, he said he was "disappointed in our choices for president" and did not support Mr Trump any more.
He said: "I can no longer look past the pattern of behaviour and comments that have been made by Donald Trump."
The governor of Alabama, who had previously endorsed John Kasich, said: "I cannot and will not vote for Donald Trump."
The governor of Nevada said: "This video exposed not just words, but now an established pattern which I find to be repulsive and unacceptable for a candidate for President of the United States."
The governor of South Dakota said: "Enough is enough. Donald Trump should withdraw in favour of Governor Mike Pence. This election is too important."
The Minnesota congressman said the comments were "disgusting and offensive" and he would not vote for Mr Trump.
A senator in Alaska, he said: "Keeping Republicans in the Senate majority is critical to the economic and national security of Alaska and America.
"As for the White House, Donald Trump should step aside. I will support Mike Pence for president."
An Alabama congressman, he said: "There are absolutely no circumstances under which it would ever be appropriate to speak of women in such a way.
"It is now clear Donald Trump is not fit to be president of the United States and cannot defeat Hillary Clinton."
The Michigan congressman said Mr Trump should "carefully consider stepping aside from the ticket".
He said: "I urge him to think about our country over his own candidacy."
An Illinois congressman, he took the rare step of referencing his sons when explaining why he would not support Mr Trump any more. Many others had referenced their female relatives.
In a statement, he said: "As parents of a teenage daughter and teen twin boys, my wife and I teach them to respect women and that they will be judged by their words and actions.
"The abhorrent comments made by Donald Trump are inexcusable and go directly against what I've been doing in Washington to combat assaults on college campuses. Because of this, I am rescinding my support for Donald Trump and asking to have my name removed from his agriculture advisory committee."
The Florida congressman originally supported Marco Rubio for the presidential nomination, but then called for party unity behind Mr Trump.
Following the publication of the tape, he said: "My greatest responsibility in life is to try and be a good husband and father. If I support him for president, I will be telling my boys that I think it's okay to treat women like objects - and I'll have failed as a dad. Therefore, I can no longer support Donald Trump for president and will not be voting for him or Hillary Clinton."
The Missouri congresswoman said: "As a strong and vocal advocate for victims of sex trafficking and assault, I must be true to those survivors and myself and condemn the predatory and reprehensible comments of Donald Trump."
The New Hampshire senator, who is running for re-election in November, withdrew her support from Mr Trump and said she would write Mike Pence's name on her ballot paper instead.
She said: "I'm a mom and an American first, and I cannot and will not support a candidate for president who brags about degrading and assaulting women.
Also facing a re-election battle, this Ohio senator said: "While I continue to respect those who still support Donald Trump, I can no longer support him."
Mr Glenn is running for the Senate. From Colorado, he asked Mr Trump to "do the honourable, selfless thing" and step aside to allow Mr Pence to run for President.
In a statement, he wrote: "If Trump is truly committed to making America great again, then this is the only way forward.
"As a father, as a Christian, and as a Republican, I believe that we simply cannot tolerate a nominee who speaks this way about women."
The chairman of the Republican Conference and senator for South Dakota said: "Donald Trump should withdraw and Mike Pence should be our nominee effective immediately."
The former secretary of education served under Ronald Reagan. He is now a conservative commentator and it was only in August that he said Republicans who opposed Mr Trump "suffer from a terrible case of moral superiority and put their own vanity and taste above the interest of the country".
But after seeing the video, he too called on Mr Trump to stand down, saying: "It's over. I hate to say it, but it's over."
"It's a shame, a crying shame, but he can't win," he said.
The House speaker, who endorsed Mr Trump just a few weeks ago, has withdrawn an invitation for the presidential candidate to attend the Republican Fall Fest in his home state of Wisconsin this weekend.
Mr Ryan has not pulled his endorsement, but it does pack a powerful punch for a candidate to be uninvited from an event just one month before the election.
Mr Ryan said: "I am sickened by what I heard today. Women are to be championed and revered, not objectified. I hope Mr Trump treats this situation with the seriousness it deserves and works to demonstrate to the country that he has greater respect for women than this clip suggests. In the meantime, he is no longer attending tomorrow's event in Wisconsin."
The Senate majority leader said the comments were "repugnant", adding that Mr Trump "needs to apologise directly to women and girls everywhere".
He also said that Mr Trump should "take full responsibility for the utter lack of respect for women shown in his comments on that tape".
The Utah state house speaker said he hoped for an apology from Mr Trump but did not rescind his support.
He said: "To say I'm disappointed would be a gross understatement."
The former Florida governor, who ran unsuccessfully for president this year, tweeted: "As the grandfather of two precious girls, I find that no apology can excuse away Donald Trump's reprehensible comments degrading women."
He did not mention his cousin Billy Bush, who was Mr Trump's interlocutor in the incriminating video.
The former presidential candidate has been critical of Mr Trump in the past and said he would not vote for him.
After the video emerged, he tweeted that the comments the current candidate made were "vile degradations" with an impact on women and on the image of the US around the world.
He wrote: "Hitting on married women? Condoning assault? Such vile degradations demean our wives and daughters and corrupt America's face to the world."
The California congressman said: "In my career as a law enforcement professional I was confronted with and worked tirelessly to end the horrific reality of violence toward women.
"After hearing Donald Trump's inexcusable comments I was deeply disturbed & reminded of that reprehensible behaviour.
"While I've never before endorsed a Presidential candidate, I've felt compelled to strongly condemn many of Mr. Trump's previous outrageous remarks. And after serious consideration, I have decided that I cannot support either candidate for President."
The governor of New Mexico had never endorsed Mr Trump. She said: "What Trump brags about is appalling and completely unacceptable.
"No woman should ever be treated the way he claims he treated women. Unfortunately, there is a pattern of disturbing conduct and offensive rhetoric that raises serious questions about his fitness to be President.
"That's why I have withheld my support from the very beginning, and will not support him now."
The Ohio governor, who also ran for the presidency and was beaten by Mr Trump, had already said he would not vote for him. He tweeted his condemnation of Mr Trump's comments, with the words: "Make no mistake the comments were wrong and offensive. They are indefensible."
A congresswoman from Utah, where much of this wave of condemnation of Mr Trump began, she said: "For the past several months I have been one of the few who refused to endorse Donald Trump.
"I have said all along that I was still waiting for Mr Trump to demonstrate his commitment to the kinds of principles and policies the people in Utah's 4th Congressional District want in their elected leaders. Mr Trump has yet to clear that bar and his behaviour and bravado have reached a new low.
"I cannot vote for him. For the good of the party, and the country, he should step aside."
The senator for Colorado called on Mr Trump to stand down, calling him "a candidate whose flaws are beyond mere moral shortcomings and who shows a disgust for American values and a disdain for dignity unbecoming of the Presidency".
The lieutenant-governor of New Jersey had not endorsed Mr Trump, despite working closely in her home state with his prominent supporter Chris Christie.
In response to a question on Twitter, she wrote: "No apology can excuse away Mr. Trump's reprehensible comments degrading women. We're raising my three boys to be better than that."
She later said she would vote, but not for Mrs Clinton and not for Mr Trump.
The Pennsylvania congressman was another Republican politician who had declined to endorse Mr Trump, deflecting a question when asked his stance back in May. After the 2005 video surfaced, he called for Mr Trump to "end his candidacy" and said: "This sort of vile talk is appalling, it's offensive, and there's no place in public or private for it. It's simply wrong.
The Governor of Tennessee called on Mr Trump to stand aside and make way for Mr Pence. He said he would write another Republican's name on the ballot paper otherwise, and added: "Character in our leaders does matter. None of us in elected office are perfect, but the decisions that are made in the Oval Office have too many consequences to ignore the behaviour we have seen."
In a text message to his local newspaper, the Texas congressman wrote: "I never endorsed Trump and I cannot in good conscience support or vote for a man who degrades women, insults minorities and has no clear path to keep our country safe. He should step aside for a true conservative to beat Hillary Clinton."
A congressman from New York, he has previously disavowed Mr Trump's "tone and rhetoric" but after the video, told his local newspaper that "what he was talking about is tantamount to sexual assault".
Mr Katko called on Mr Trump to stand aside and let somebody else run, saying:  "I've talked repeatedly with my family and people I consider close to me. We are all roundly disgusted.
"We just all came to the same conclusion that he does not deserve support. You just can't explain this away."
A congressman for Idaho, he said: "While I've never endorsed Donald Trump, I find his recent comments about women deplorable. In my opinion, he has demonstrated that he is unfit to be president and I cannot support him."
The only woman who represents Texas in the US Congress had previously declined to endorse Mr Trump. After the video surfaced, she said: "We have heard rumours about the insensitive and vulgar things Mr. Trump says about women. But watching that video is disgusting. Mr. Trump should remove himself from consideration as Commander in Chief."
A Washington congresswoman, she said she will write in Paul Ryan's name on her ballot paper, instead of voting for Mr Trump. In an email, she wrote: ""For months I've left the door open for Donald Trump to earn my vote. That door has now slammed shut."
Donald Trump's campaign for the US presidency has been thrown into turmoil by a video carrying obscene remarks about women.