The protesters, who are made up of students and supporters of the Occupy Central pro-democracy movement, want a free choice of candidates for the vote in 2017 but Beijing has ruled that option out.
BBC News readers in Hong Kong and mainland China have been explaining why the issue has caused so much anger on the streets.
David Boyton, former police officer, Hong Kong
As a lawyer and a former Hong Kong police officer I did not initially care much for the Occupy Central movement.
I also felt greatly inconvenienced when I was stuck in Hong Kong traffic for several hours at one point because of these protests.
However, when I saw the unnecessary gassing of protesters who were not doing anything, my attitude completely changed.
This is an indication of what is to come - PRC [People's Republic of China]-style heavy-handed police tactics to deal with its own citizens.
It is no coincidence that the underlying movement promotes democracy and government accountability.
Why do the police, who are themselves citizens of Hong Kong, have to treat its own citizens with such disdain and contempt?
No doubt it will come out in the wash that, "we were just following orders". The unnecessary escalation by the Hong Kong police has changed my attitude completely.
The government and police must be made accountable; and this society must be allowed to move ahead in the way that the majority wishes.
I now fully support the protesters and I am ashamed of ever being a police officer in Hong Kong.
SK, a student at the Chinese University of Hong Kong
I spent a tense night amongst the students when it all started kicking off on Friday night.
The police just burst into a crowd that was formed by thousands of students and citizens who were outside the Hong Kong government headquarters.
Police used huge pepper sprays to attack the protesters. Students were shouting "protect the students, police please leave!" I support the student protests because Beijing's solution is unacceptable.
Hong Kong's leader, CY Leung, has ignored our demands to meet and that's why people in Hong Kong are so angry - not just us students.
Clara Me, Hong Kong
It's not that I don't sympathise with the protesters but I feel their way of going about things is too extreme.
I understand why they are doing it - but it is very naive to think if the students protest or speak loudly enough it will change things.
The Hong Kong government is under China's jurisdiction and if China doesn't agree, the Hong Kong government can't act. It's a difficult situation.
No-one wants another Tiananmen Square. People have been spreading rumours on social media of guns and tanks outside of Hong Kong and the territories - they are old pictures, it's not what is happening now.
So it makes me think some people want things to escalate. I want the protesters to calm down and go home.
Jason Liu, a civil engineer from Hong Kong
The students have been calm and peaceful.  Some brought food and water with them, and the protest has been well organised.
The students have been lying peacefully on the ground. The police seized a protester's loudspeaker and this is what caused the first disturbance.
The protesters were not happy about it and they tried to stop the vehicle that was taking it away.
The police are trying to persuade protesters to go away. The police increased the number of barricades to stop people going into the government square.
I believe they will escalate their actions in order to clean up the square. But the protesters have been sitting peacefully.
I've never seen this happen in Hong Kong in my life.
I want the world to know that we are peaceful; no looting or vandalism, but police still treat us as if we are rioting.
We only use umbrellas to protect ourselves from the tear gas.
Ben Sun, sales manager, Shenzhen, China
I don't support the protesters, especially those who are secondary school students. They are the ones in society who need to be protected from politics, not involved in it.
1 October is National Day. Whatever China looks like and feels like in everyone's heart, it is Mother's Day too.
No-one hates their mother even if she is ugly, poor or unwise. To protest on Mother's Day is no different to a rebellious child misbehaving.
People in Hong Kong are full of idealism. Do the university students even know what they are protesting about? I think they are too young.
Just like an older brother to a younger brother, I would suggest, please talk to your parents and respect their decision.
When you can raise children yourself and support your parents, then you will really understand.
Jessica Yeung, professor, Hong Kong
I agree with the protesters.
Civic Square is supposed to be a public space but a fence three metres high was erected around it only a few months ago. I think the government was preparing for protests.
I saw thousands of people gathering there. It was very moving. People were coming with food, drink, facemasks, raincoats and cling film - so as to protect against the pepper spray.
Beijing is being increasingly manipulative and controlling when it comes to Hong Kong's legislation.
There is a huge cultural difference and I don't think Hong Kong's legislation reflects the view of the residents to Beijing. This has caused a lot of misunderstanding.
The present government is responsible to a large extent for what is happening. They've blocked communications between democrats and Beijing.
Andy Bugden, Shenzhen, China
I don't know what the protesters expect to achieve.  I think Hong Kong people look down on mainlanders.
In about 30 years Hong Kong will return obediently to the mainland under one system.
They need to deal with reality. If they work with China then all will be fine - work against and they will find that China is stronger.
Hong Kong people are not going to get a choice that they want. A stable government is good for business.
The people of Hong Kong have to see the long-term view. They won't have democracy until China has democracy.
Tens of thousands of protestors have filled the streets of central Hong Kong demanding that China withdraws rules that allow it to vet candidates for the next leadership election.