The twenty-seven remaining European Union countries have unanimously adopted guidelines for negotiating Britain's withdrawal.
European Council president Donald Tusk said the summit in Brussels had approved a firm and fair mandate for setting the terms of Brexit.
He called on the UK to come up with a "serious response" on what would happen to EU citizens in Britain after Brexit.
"We need guarantees," he said.
The rights of EU citizens to live, work and study in the UK was one of three topics they wanted dealt with in the first phase of Brexit talks.
Negotiations will start after the UK election on 8 June.
Mr Tusk put citizens' rights centre stage at a news conference after EU leaders - minus UK PM Theresa May - agreed on the guidelines in a matter of minutes.
"Over the past weeks, we have repeatedly heard from our British friends, also during my visit in London, that they are ready to agree on this issue quickly," he said.
"But I would like to state very clearly that we need real guarantees for our people to live, work and study in the UK.
"The same goes for the Brits," living on the European continent, he said.
UK citizens living in EU countries and non-UK EU citizens living in Britain are estimated at 4.5 million.
The EU's negotiating guidelines, first proposed by Mr Tusk in March, list citizens' residency rights, settling Britain's financial commitments to the EU and avoiding a "hard" border between the Irish Republic and Northern Ireland as the three top issues needing agreement in what were termed "separation talks".
Only once "sufficient progress" was made on these topics could talks touch on the UK's future relationship, including any trade deal, with the EU.
However, the UK government has pushed for parallel negotiations on trade.
'We are ready... we are together'
Speaking after the summit, European Commission President Jean-Claude Juncker again stressed that separation talks could not run in parallel with talks on a future trade deal with the UK, backing the line taken by German Chancellor Angela Merkel as she arrived in Brussels.
EU officials said leaders burst into applause as the negotiating stance was waved through at the summit.
EU leaders and officials were keen to stress the EU's unified position on Brexit. Chief negotiator, Michel Barnier, said: "We are ready ... we are together."
Speaking earlier, French President François Hollande said there would inevitably be "a price and a cost for the UK - it's the choice that was made".
"We must not be punitive, but at the same time it's clear that Europe knows how to defend its interests, and that Britain will have a weaker position in the future outside Europe, than it has today within Europe."
On the issue of the UK's financial obligations, EU officials estimated that Britain faced a bill of €60bn because of EU budget rules. UK politicians have said the government would not pay that much.
UK Brexit Secretary David Davis said both sides wanted the negotiations to be conducted with goodwill.
But he said "there is no doubt that these negotiations are the most complex the UK has faced in our lifetimes. They will be tough and, at times even confrontational".
- 29 April - EU leaders (excluding the UK) meet in Brussels to adopt Brexit negotiating guidelines
- 7 May - French voters decide between Emmanuel Macron and Marine Le Pen as their next president
- 8 June - UK parliamentary election - Brexit talks to start soon after the vote
- 24 September - German parliamentary election, with Mrs Merkel seeking a fourth term
- 29 March 2019 - Deadline for ending talks on UK exit terms (any extension requires agreement of all member states)
- May or June 2019 - European Parliament election (without UK)
- Ratification - Any Brexit deal requires ratification by all EU's national parliaments and European Parliament