Three million EU nationals living in the UK will be allowed to remain indefinitely after Brexit provided UK citizens in Europe are give the same treatment, the prime minister has said.
Theresa May outlined what she called a "fair and serious offer" at a dinner ahead of the second day of the European Council's two-day summit in Brussels.
The summit is Mrs May's first since the general election that cost her the Tory's parliamentary majority.
But she won a cautious welcome from EU leaders for the plan, which would allow EU nationals who have been in the UK more than five years to claim a new "settled status", entitling them to similar rights as full British citizens.
Those who have been in the UK for a shorter time would be able to stay on until they hit the five-year threshold for settled status.
Others who arrive after a cut-off date will be given a "grace period" - expected to be two years - to regularise their immigration status.
- What is Mrs May's proposal for EU citizens in Britain?
But Mrs May made clear that the proposals would be adopted only if there was a reciprocal agreement for UK citizens living in the remaining 27 EU states.
German Chancellor Angela Merkel said the proposals represented "a good start", but cautioned that there were "many, many other issues" before Britain could reach agreement on a withdrawal deal.
The plan received a more frosty reception at home, with Labour's shadow Brexit secretary Keir Starmer calling it "too little too late".
Mayor of London Sadiq Khan, said Mrs May's plan "does not come close to fully guaranteeing the rights of EU nationals living in the UK".
And Liberal Democrat leader Tim Farron said the proposals "leave millions of people still facing unanswered questions over their futures here".
There are plenty of questions remaining and further details are expected in a government paper on Monday.
But a key area of disagreement is expected to be over jurisdiction.
The EU proposals would give the European Court of Justice "full jurisdiction" for as long as citizens' rights remain protected under the withdrawal agreement.
But Mrs May told her fellow leaders: "The commitment that we make to EU citizens will be enshrined in UK law and will be enforced through our highly respected courts."
Also at the summit the remaining 27 countries agreed a process to move EU banking and medicines regulators out of the UK after Brexit.