Theresa May said there has been a "constructive" start to the Brexit talks as she prepared to set out proposals to break the logjam over the status of expat citizens following UK withdrawal from the EU.
Arriving at the European Council summit in Brussels, the Prime Minister reaffirmed her intention that the future of the 3.2 million EU nationals in the UK and 1.2 million UK citizens in the EU should be one of the first issues to be resolved.
"We will be going into negotiations. Those have started constructively," she told reporters.
"What I am going to be setting out today is clearly how the United Kingdom proposes to protect the rights of EU citizens living in the UK and see the rights of UK citizens living in Europe protected.
"That's been an important issue. We've wanted it to be one of the early issues to be considered in the negotiations That is now the case. That work is starting."
Mrs May, appearing at the summit just days after talks opened between Brexit Secretary David Davis and the EU's chief negotiator, Michel Barnier, said there would be other issues on the agenda, including counter-terrorism.
"One of the things that I will be calling on with other European leaders today is that we do more working together to ensure that we stop the spread of extremism on line, that we prevent terrorists from having a safe space online and we keep our citizens safe," she said.
Ahead of her visit, Foreign Secretary Boris Johnson said he hoped leaders of the 27 other nations would match her "generous" proposals to guarantee the rights of EU nationals with a similar offer to the British expats on the Continent.
"What the Prime Minister will be able to do on Thursday is set out, as she has tried actually to do several times, her instinct to be generous about the 3.2 million EU citizens who are living here and hoping that there will be reciprocal, corresponding generosity towards the one million UK nationals in the rest of the EU," he said.
"I think she's got a great offer to make and I hope it will go down well."
The 27 remaining members will also hear a report from Mr Barnier on his talks with Mr Davis earlier this week, when the Brexit Secretary accepted a European timetable to put off talks on a future trade relationship until progress has been made on the terms of the divorce.
In a two-day summit whose agenda is formally dominated by immigration, security and the economy, Mrs May will also brief her counterparts on the UK's commitment to a new £75 million plan designed to stem the flow of illegal migrants from Africa to Europe.
The three-year programme will offer humanitarian support, including food and water, to would-be migrants on the perilous transit routes from the Horn of Africa and western Africa through countries including Niger, Egypt and Libya.
And migrants who find themselves stranded and destitute along the routes will be offered assistance to return home.
While any assisted returns will be voluntary, it is thought that many individuals could take advantage of the scheme as a way of escaping the gruelling conditions of the transit routes, which expose them to the risk of death, violence, forced labour and exploitation at the hands of people-traffickers before they even reach Mediterranean Sea ports.
A record 4,576 people are thought to have died or been recorded missing while attempting the risky central Mediterranean Sea crossing during 2016.
Some 181,000 people arrived in Italy via this route last year, with a greater number expected in 2017.
International Development Secretary Priti Patel said: "The UK has been at the forefront of responding to the migration crisis and our work to date has helped reduce total migrant numbers to Europe since 2015.
"But, worryingly, more and more people are now using the incredibly dangerous central Mediterranean route.
"This new UK support will provide desperately needed aid and protection to tens of thousands of the world's most vulnerable.
"But, critically, it will also make clear the massive risks involved at every stage of this route and provide alternatives so those who change their minds can return home.
"As well as saving lives, this will provide vulnerable people with meaningful alternatives to the treacherous crossings into Europe.
"Building on our existing work to deal with the root causes of migration, this approach is about using our aid in a smart way, as part of a co-ordinated approach across Government, to provide protection to people who need it and serve Britain's national interests."
German Chancellor Angela Merkel said building a future for the remaining 27 EU member states took priority over the negotiations with the UK.
Arriving at the Brussels summit, she said: "I want to say clearly that, for me, shaping the future for the 27 takes priority over the negotiations with Great Britain over withdrawal.
"Naturally we will conduct these negotiations quickly and we will conduct them intensively. We will do everything to ensure that - as has been successfully done so far - the 27 states stick together.
"We want this negotiation to take place in a good spirit. We know that we will want to work with Great Britain later. But the clear focus must be on the future of the 27, so that we have the best results."
The president of the European Parliament, Antonio Tajani, said: "For us, it is important to conclude Brexit with a good solution for both the EU and the UK. I think we need to have a framework for separation before the end of 2018, then start on negotiation for the future, for the day after the separation between the EU and UK."
Asked whether he believed Brexit would definitely happen, Mr Tajani said: "On this, I am optimistic. We need to work. We have a good negotiator - Michel Barnier is a very good man, very experienced. He knows the UK. I think we will achieve a good result."
Speaking ahead of the summit, European Council president Donald Tusk quoted John Lennon to indicate that he believes the door remains open for the UK to stay in the EU despite the start of Brexit talks.
"Some of my British friends have even asked me whether Brexit could be reversed and whether I could imagine an outcome where the UK stays part of the EU," said Mr Tusk.
"I told them that, in fact, the European Union was built on dreams that seemed impossible to achieve. So, who knows - 'You may say I'm a dreamer, but I'm not the only one'."
And France's new president, Emmanuel Macron, restated his view that the UK could perform a U-turn on Brexit, and acknowledged the need to address the problems caused by free movement of people - one of the triggers for the Leave vote.
He told the Guardian: "The door is open until the moment you walk through it. It's not up to me to say it's closed.
"But from the moment things are engaged with a timescale and an objective, it's very hard to go back, we can't lie to ourselves."
Mr Macron said that for the EU to achieve greater support from its citizens, rules on movement of workers needed to be tightened to make it harder for wages to be undercut by migrant labour or outsourcing to other countries in the bloc.
He said: "Let's not get this wrong. The great defenders of this ultra-economically liberal and unbalanced Europe - the UK - came crashing down on this.
"What did Brexit play on? On workers from Eastern Europe who came to take British jobs. The defenders of the European Union lost because the British lower middle classes said 'Stop!'"
Arriving for his first European Council summit as president, Mr Macron said his priority was to talk about Europe's ambitions, policies and projects rather than "having discussions lasting days and nights about dismantling it".