The Prime Minister said the plan she outlined to a European Council summit in Brussels on Thursday would ensure that no-one living lawfully in the UK will have to leave and that EU nationals will not face the prospect of their families being split up by Brexit.
Mrs May's proposals - which are subject to reciprocal arrangements for UK citizens in the remaining 27 EU states - won a guarded welcome from fellow European leaders, but were denounced by British opposition parties as "too little too late" after a year in which EU nationals had been treated like bargaining chips.
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Arriving for the second day of the summit on the anniversary of the June 23, 2016, referendum which paved the way for Brexit, Mrs May acknowledged that elements of her plans would be contested by the EU.
"Of course, there will be details of this arrangement which will be part of the negotiation process," she said.
But she added: "I want to reassure all those EU citizens who are in the UK, who have made their lives and homes in the UK, that no-one will have to leave, we won't be seeing families split apart.
"This is a fair and serious offer. I want to give those EU citizens in the UK certainty about the future of their lives, but I also want to see that certainty given to U citizens who are living in the EU."
Mrs May's plan would allow EU nationals who have been in the UK more than five years to claim a new "settled status" entitling them to the same rights as full British citizens to healthcare, education, welfare and pensions.
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, while others who arrive after a cut-off date will be given a "grace period" - expected to be two years - to regularise their immigration status.
The cut-off date for entitlement to apply is yet to be set, but will come between the day when Britain formally notified Brussels of its intention to quit on March 29 2017 and the day when it finally leaves, expected to be March 29 2019.
This conflicts with the formal proposals set out by the EU, which would grant residency rights all the way up to the final date of withdrawal.
Brussels is also demanding a role for the European Commission to monitor and the ECJ to enforce the rights of EU citizens who remain after Brexit.
Mrs May set her face against ECJ involvement as she briefed her fellow leaders over dinner on Thursday, telling them: "The commitment that we make to EU citizens will be enshrined in UK law and will be enforced through our highly respected courts."
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.
Dutch Prime Minister Mark Rutte said there were "thousands of questions to ask" about Mrs May's proposals, and Austrian Chancellor Christian Kern described them as "a first step" which did not cover the situations of many EU citizens in the UK.
Mrs May also promised that the system will be streamlined, doing away with the 85-page permanent residency application form which has provoked loud complaints.
It is thought the UK is reserving the option of setting an early cut-off for residency rights in case there is a late surge of migrants arriving as Brexit approaches.
But the introduction of a "grace period" raises the prospect that large numbers arriving during withdrawal negotiations may be allowed to remain.
And the outline deal leaves questions unanswered over whether individuals with settled status will be permitted to bring in children or spouses and whether the new status will be subject to conditions other than length of residency. Further details are expected in a Government paper on Monday.
Shadow Brexit secretary Sir Keir Starmer said: "The Prime Minister's offer is too little too late and falls far short of the full and unilateral guarantee Labour would make.
"We believe there must be a clear commitment that there will be no change to the status of EU nationals in the UK. This is not only the right thing to do, but it will also help deliver a reciprocal agreement for the 1.2 million UK nationals living in the EU."
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".
Liberal Democrat leader Tim Farron said the proposals "leave millions of people still facing unanswered questions over their futures here".
Green Party co-leader Caroline Lucas said: "This doesn't end uncertainty facing EU citizens who have made their home here. Shameful Theresa May still using them as bargaining chips."
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