The EU and Turkey say they have agreed the broad principles of a plan to ease the migration crisis.
Under the plan discussed in Brussels, all migrants arriving in Greece from Turkey would be returned.
For each Syrian sent back, a Syrian already in Turkey would be resettled in the EU. Turkey would also get extra funding and progress on EU integration.
However, the deal has not been finalised and talks will continue ahead of an EU meeting on 17-18 March.
Europe is facing its biggest refugee crisis since World War Two. Last year, more than a million people entered the EU illegally by boat, mainly going from Turkey to Greece.
Read more about the migrant crisis
- Crisis explained in seven charts
- How different countries have been affected
- Key migrant crisis questions answered
- Have previous EU migrant deals delivered?
- Disquiet over deal with Turkey
Most of them were Syrian, fleeing the country's four-year civil war. Another 2.7 million Syrian refugees are currently in Turkey.
What's in the proposal?
The EU heads said "bold moves" were needed to tackle the crisis, and made the following proposals:
- All new irregular migrants crossing from Turkey to the Greek islands will be returned to Turkey, with the EU meeting the costs. Irregular migrants means all those outside normal transit procedures, ie without documentation. The term "illegal migration" usually refers to people smuggling
- In exchange for every returned Syrian, one Syrian from Turkey will be resettled in the EU
- Plans to ease access to the EU for Turkish citizens will be speeded up, with a view to allowing visa-free travel by June 2016
- EU payment of €3bn ($3.3bn; £2.2bn) promised in October will be speeded up, and a decision will be made on additional funding to help Turkey deal with the crisis. Turkey reportedly asked for EU aid to be increased to €6bn
- Preparations will be made for a decision on the opening of new chapters in talks on EU membership for Turkey
How have leaders reacted?
European Council President Donald Tusk insisted the leaders at the summit had made a "breakthrough", and he was hopeful of concluding the deal in the next week.
He said the progress sent "a very clear message that the days of irregular migration to Europe are over".
However, German Chancellor Angela Merkel was more circumspect, saying: "It is a breakthrough if it becomes reality."
Turkish Prime Minister Ahmet Davutoglu said Turkey had taken a "game-changing" decision "to discourage illegal migration, to prevent human smugglers, to help people who want to come to Europe through encouraging legal migration".
The BBC's Chris Morris in Brussels says that, although this new initiative is bold, it could spark fierce argument and its implementation will not be easy.
Can the return system work?
The system spelled out to the BBC by EU Commission spokesperson for migration Natasha Bertaud would see all migrants rescued in Greek waters taken to a Greek island for screening.
All economic migrants would then be returned to Turkey where they would be screened again and "if they have no right to international protection" (which currently covers only Syrians) sent back to their country of origin.
All migrants rescued by Nato in Turkish waters would be taken back to Turkey, which would decide if their status.
Serious questions remain.
What will happen to the thousands of migrants already in Greece, which has struggled to shelter and register them?
What is the legal status of returning migrants to Turkey? As it is not a full member of the Geneva Convention, could the system be challenged in courts? In addition, only one member of the EU – Bulgaria – currently considers Turkey a safe country of return.
The UN's refugee agency has already voiced concern about the "blanket return" of people without clarity on their legal status.
The one-in, one out system also only applies to Syrians. What will happen to all the other migrants returned to Turkey? Again the legality of their return must be considered, as must Turkey's capability to return them to their countries of origin.
The biggest problem, though, will be the migrants themselves – having risked their lives and invested much of their money, will they not simply try other routes? The migrants in the Calais camp known as "the Jungle" have not been known to give up on their attempts to reach the UK.
As for resettlement, there is major opposition among some EU members for compulsory migrant quotas.
What are the other obstacles?
Hungary's anti-migration PM Viktor Orban has already indicated he may veto the resettlement deal.
Turkey's bid for EU membership. A long and thorny issue, not helped by the recent press-freedom wrangle over the court-ordered seizure of the opposition Zaman newspaper. Given all the hurdles, though, this is not a pressing concern.
More problematic is Turkey's request for visa-free access for all its citizens to the EU's Schengen zone, which it hopes to achieve by June. This may draw a lot of opposition.
The future of Schengen – which allows passport-free travel in a 26-nation zone – is already in doubt, given that eight of its members have introduced temporary border controls.
A note on terminology: The BBC uses the term migrant to refer to all people on the move who have yet to complete the legal process of claiming asylum. This group includes people fleeing war-torn countries such as Syria, who are likely to be granted refugee status, as well as people who are seeking jobs and better lives, who governments are likely to rule are economic migrants.