David Cameron said politics had lost a man of "immense ability", while Nick Clegg said his opposition to the Iraq war had been "enormously courageous".
Mr Kennedy, who led his party for over six years, took it to its best election result in 2005 but battled alcoholism.
No cause of death has been given but police said it was not suspicious.
Mr Kennedy, who lost his seat in last month's election, died at his home in Fort William on Monday. His family said they were devastated to lose a "fine man and loving father".
Under Mr Kennedy's leadership, the Lib Dems won a record 62 seats in 2005 but he resigned eight months after the election after revealing he had been receiving treatment for a long-standing drink problem.
Charles Kennedy: 1959-2015
By Nick Robinson, BBC political editor
Charles Kennedy left a mark on British politics. The man who took his party to its electoral peak, he was the only UK party leader to warn the country of the perils of invading Iraq when Labour and the Conservatives were uniting to support it.
He was also the only Liberal Democrat MP who could not bring himself to vote to form a coalition with the Conservatives.
But British politics also left its mark on him. Elected at the age of just 23, politics and the House of Commons became his life whilst alcohol was his friend, his prop and his curse.
Read Nick's thoughts
John Pienaar's memories
The Charles Kennedy story
Charles Kennedy's life in pictures
Charles Kennedy in his own words
Scottish tributes
Mr Clegg, who will step down as Lib Dem leader next month, said that on a good day Mr Kennedy had "more political talent in his little finger than the rest of us put together".
"Charles devoted his life to public service, yet he had an unusual gift for speaking about politics with humour and humility which touched people well beyond the world of politics," he said.
"He was one of the most gentle and unflappable politicians I have ever known, yet he was immensely courageous too - not least when he spoke for the country against the invasion of Iraq."
Mr Kennedy's family said in a statement: "It is with great sadness, and an enormous sense of shock, that we announce the death of Charles Kennedy.
"We are obviously devastated at the loss. Charles was a fine man, a talented politician, and a loving father to his young son."
A Police Scotland spokesman said: "Police officers attended an address at Fort William on Monday, June 1 to reports of the sudden death of a 55-year-old man. Police were notified by ambulance service personnel. There are no suspicious circumstances."
Mr Kennedy's political career began in the Social Democratic Party and he became the youngest MP of the time when he won the Ross, Cromarty and Skye seat in 1983 at the age of 23.
At first he was SDP spokesman on social security, Scotland and health and when most of his party merged with the Liberals to form the Lib Dems in 1988, he continued to hold a series of frontbench posts.
He took over the Liberal Democrat leadership from Paddy Ashdown in 1999.
During the 1990s, Mr Kennedy built up his profile through a series of TV appearances, earning him the nickname, which he hated, of Chatshow Charlie.
His 2002 marriage to Camelot public relations executive Sarah Gurling was seen by many in the party as a sign he was settling down.
The birth of his son in 2005 was seen as a further sign that the hard-partying Kennedy - one commentator had dubbed him "Jock the lad" - was being transformed into a family man, although he and his wife split up in 2010.
After his resignation, Mr Kennedy maintained a lower profile. He did not play any role in the coalition government, having voted against his party's decision to enter an alliance with the Conservatives.
Writing in May 2010, he said the tie-up with the Conservatives drove "a strategic coach and horses through the long-nurtured 'realignment of the centre-left' to which leaders in the Liberal tradition, this one included, have all subscribed" since the 1950s.
'I rated his intellect, empathy, humanity and his humour'
By Brian Taylor, BBC political editor, Scotland
When a politician dies, the response is occasionally formulaic. Life of service, dedication and duty, will not see the like again.
With Charles Kennedy, the respect is genuine - and the sympathy enormous. After all, this is a man who lost his father (who died in April), who lost his seat in May and who has now lost his life in June.
I first encountered Charles when he entered the Commons as the youngest MP in 1983, making him the Baby of the House. I was a few years older, a correspondent at Westminster for the Press and Journal.
As the P&J reach encompasses the Highlands, Charles was an important contact for me. But he was more. He was a chum. I rated his intellect, his empathy, his humanity and his humour.
He always struck me as slightly removed from politics. By that, I mean politics as routinely practised, politics in conventional form.
Read Brian's full thoughts
Latest tributes from Scotland
As tributes to Mr Kennedy poured in from across the political spectrum, Prime Minister David Cameron said his death was a "tragic loss" for his family and for public life in general.
"He was someone of immense ability," he said. "It's not that often in politics that someone comes along with brains, talent, wit and bags of humanity and Charles had all of those things."
Acting Labour leader Harriet Harman said his death was a huge loss, describing Mr Kennedy as "a delightful person with a great intellect".
Tony Blair, who clashed with Mr Kennedy over the former Labour prime minister's decision to take the UK to war in Iraq, said he had the greatest of respect for the man who had entered Parliament at the same time as him, in 1983.
65 MPs
Liberal Democrats saw their best-ever result in 2005 while Charles Kennedy was party leader
32 years as MP for the Scottish constituency of Ross, Skye and Lochaber
23 years-old when first elected in 1983
8 general elections fought as a member of the SDP and then the Liberal Democrats
"Charles's death is an absolute tragedy. He was throughout his time a lovely, genuine and deeply committed public servant," he said.
"As leader of the Liberal Democrats, we worked closely together, and he was always great company, with a lively and inventive mind."
Scotland's First Minister Nicola Sturgeon tweeted: "Sad beyond words to hear the news about Charlie Kennedy. A lovely man and one of the most talented politicians of his time. Gone too soon."
Lord Ashdown told the BBC that Mr Kennedy "had his demons, we all have our demons, but on form and when he was on song Charles was the best of all of us".
Lib Dem peer Baroness Williams described Mr Kennedy as a "wonderful" and "hugely generous" man who had a "big view" of politics.
"He saw the whole world as a place where you could work for social justice, you could work for greater equality, you could work for greater opportunity for people like himself. He came from a pretty humble background," she told the BBC.
Reflecting on Mr Kennedy's battle with alcohol, former Lib Dem MP Sir Malcolm Bruce said the party did not handle his exit as leader well, telling the BBC there were times when "we hoped he was coping with it but he wasn't always".
And Alastair Campbell, the former adviser to Tony Blair who has tackled his own problems with alcohol, said the two men - who became close friends - faced a "shared enemy".
"Alcoholism is a disease and people should reflect on that," he told BBC Radio 5 live. "Charles struggled with it and I think it's harder to struggle with it if you're in the public eye - and sometimes he was winning the struggle and sometimes he was losing the struggle.
"But through it all he was basically the same guy - very warm, very funny, very engaging, very good with people and very principled."
Tributes will be paid to Mr Kennedy in Parliament on Wednesday after Prime Minister's Questions, Commons Speaker John Bercow said, remarking that he had been a "great parliamentarian".
Colleagues, friends and political adversaries have paid tribute to former Liberal Democrat leader Charles Kennedy after his sudden death aged 55.