Hong Kong officials have ended their first round of talks with students, with the government's chief negotiator saying she hoped for further meetings with protest leaders.
Students at the talks reiterated their demand for an unrestricted choice of candidates in the election for the territory's chief executive in 2017.
But both Hong Kong and Beijing officials have said this is impossible.
Protests have blocked key parts of the city, although numbers have fallen.
The BBC's Juliana Liu in Hong Kong says that although demonstrators know the chances of getting what they want are almost zero, they are staying on the streets to show authorities that the struggle for democratic reform is a long-term fight.
The government's negotiation team was led by the city's most senior civil servant, Chief Secretary Carrie Lam, and the students were represented by five leaders.
Meetings have been called off twice in recent weeks.
The talks, which began at 18:00 local time (11:00 BST), focused on the students' demands that China reconsiders its ruling on how candidates for elections will be chosen.
Student leaders reiterated their position that they want Hong Kong leadership elections to be more democratic - a demand rejected by Ms Lam.
"As far as their position is concerned I'm afraid we can only agree to disagree," she said.
Alex Chow, secretary general of the Hong Kong Federation of Students, said the government's stance was "vague".
"We would say that the government needs to further explain it in front of the public," he added.
At the scene: Juliana Liu, BBC News, Hong Kong
In a photo call before the talks began, the differences between the two sides could not be more stark.
On one side sat the protest leaders - one woman and four men, all in their early-20s and wearing jeans and black T-shirts with the words Freedom Now written in English.
Across from them sat another group - also consisting of one woman and four men, but decades older, wearing business attire and representing the Hong Kong government.
Students versus civil servants. Youth versus middle age. Idealism versus pragmatism.
In terms of rhetoric, the students were more passionate and florid, appealing to the public with their arguments about freedom and democracy.
The government officials tended to dwell more on legalities.
But, in the end, little was agreed and the government has expressed hopes of further talks.
The meeting came a day after Chief Executive CY Leung repeated his objections to the protesters' demands, saying they would result in populist policies.
"If it's entirely a numbers game and numeric representation, then obviously you would be talking to half of the people in Hong Kong who earn less than $1,800 (£1,110) a month," he told reporters.
"Then you would end up with that kind of politics and policies."
Mr Leung said problems such as the lack of social mobility and unaffordable housing were "not acceptable", and the government needed to do more to solve them.
But he argued that Beijing's position, under which candidates will be screened by a "broadly representative" nominating committee before they go to an open election, was better.
He pointed to the fact that his own appointment in 2012 had to be endorsed by a 1,200-member committee which was made up of people from various sectors of society and professions.
Mr Leung said the make-up of the nomination committee might offer room for negotiation.
Hong Kong democracy timeline
- 1984: Britain and China sign an agreement where Hong Kong is guaranteed "a high degree of autonomy, except in foreign and defence affairs" for 50 years following the handover in 1997.
- June-July 2014: Pro-democracy activists hold an unofficial referendum on political reform and a large rally. This is followed by protests by pro-Beijing activists.
- 31 August 2014: China says it will allow direct elections in 2017, but voters will only be able to choose from a list of pre-approved candidates. Activists stage protests.
- 22 September 2014: Student groups launch a week-long boycott of classes in protest.
- 28 September 2014: Occupy Central and student protests join forces and take over central Hong Kong
- 2 October 2014: Chief Executive CY Leung refuses demands for his resignation, offers talks with government. Student leaders later accept the offer.
- 2017: Direct elections for chief executive due to take place
The protests drew tens of thousands to the streets at their peak, in a major challenge to Beijing's rule. A few hundred activists still remain entrenched at Mong Kok, Admiralty and Causeway Bay.
On Monday police warned demonstrators to stay away from one of the protest sites, in Mong Kok, saying it was on the verge of "turning into a riot".
Protesters have disputed this and say it is still peaceful.
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