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"While May was almost uniquely ill-equipped to be the negotiator we needed, the truth is she was given an impossible job," says Green MP and former leader of the party Caroline Lucas.

"You can’t achieve a hard Brexit and avoid a hard border in Northern Ireland.

"No new PM can achieve it either."

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The highs and lows of Theresa May's time in office, explained in six charts.

International Development Secretary Rory Stewart - who has said he will run as next Tory leader - also calls Mrs May "dignified", a word several ministers have used.

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DUP leader Arlene Foster says her party "enjoyed a respectful and courteous relationship" with Theresa May despite "differences in our approach, particularly on Brexit".

She thanks the prime minister for "her willingness to recognise Northern Ireland's need for additional resources".

"I pay tribute to her selfless service in the interests of the United Kingdom," she adds.

In a message posted on Twitter, Dutch Prime Minister Mark Rutte said he has just conveyed his thanks and respect to Theresa May.

He added that the "agreement reached between the EU and the United Kingdom for an ordered Brexit remains on the table".

Mrs May came into office with a positive net favourability rating, according to YouGov, external. But it fell and kept falling straight after she called a snap election in 2017.

Since the start of this year, it's been hitting new lows with the repeated rejection of her Brexit deal in the Commons. The latest figure, from earlier this month, had her at -49%.

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Leader of the Scottish Conservatives Ruth Davidson says the prime minister "always put country before party".

"Her time in office has been characterised by the hard work, resilience, quiet dignity and attention to detail for which she is known," she says.

Ms Davidson particularly praises the prime minister for "opposing the SNP's call for an immediate second independence referendum in 2017".

She also says Theresa May, as Britain's second female prime minister, has been "a role model for girls and women".

Nigel Farage, former UKIP leader who now leads The Brexit Party, has tweeted:

Last night, before Mrs May's resignation, European Commission President Jean-Claude Juncker gave an interview to German public broadcaster ARD.

Asked about the prospect of Mrs May's departure, Mr Juncker responded: "How could anybody else achieve what she couldn’t?"

He added: "If you tell people for 40 or 45 years 'we’re in it, but not really in it', we are part-time Europeans and we don’t like these full-time Europeans, then you should not be surprised if people follow simple slogans once they’re asked to vote in a referendum."

Image source, Reuters

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Lib Dem leader Vince Cable has responded to the prime minister's resignation:

“The prime minister is right to recognise that her administration has reached the end of the road," he says. “Sadly her compromises through the last three years have too often been with the right-wing of her own party, rather than about bringing the country together. “Conservative Party interest has always trumped national interest, and yet Conservative MPs continue to demand an ever more extreme Brexit policy. “The best and only option remains to take Brexit back to the people.

"I believe the public would now choose to stop Brexit, and stop the immeasurable damage the process has done to the United Kingdom."

German Chancellor Angela Merkel has noted Mrs May's decision "with respect", a spokesman said.

"The chancellor has always worked well - and with trust - with Mrs May and will continue to do so for as long as she is in office."

The latest developments in the UK are expected to be discussed at an EU Council meeting on Tuesday, the spokesman added.

Meanwhile, France's Emmanuel Macron sent a "personal message of support" to Mrs May after she resigned, Reuters reports., external

But France wants "speedy clarification" on Brexit, the report adds.

During her speech Theresa May said she was "proud of the progress" her government had made.

She highlighted what she believed to be the successes of her government:

• A falling national debt
• A modern industrial strategy - which she said focused on ensuring that the good jobs of the future will be created in communities across the whole country, not just in London and the South East

• Building more homes and helping first-time buyers onto the housing ladder
• Protecting the environment, eliminating plastic waste, tackling climate change and improving air quality
• Putting proper funding for mental health at the heart of our NHS long term plan
• Ending the postcode lottery for victims of domestic abuse
• Introducing a race disparity audit and gender pay reporting
• Setting up an independent inquiry into the tragedy at Grenfell Tower

You can read Mrs May's resignation statement in full here.

Or click here to watch her speech.

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BBC Radio 5 Live

James Landale
BBC diplomatic correspondent

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Theresa May entered Downing Street in 2016 determined to deliver Brexit. In the end, her premiership was consumed by it.

Her decision to pursue what was seen as a hard Brexit outside the single market and customs union divided Parliament and her party and sowed the seeds of her downfall.

This was not the plan when the newly elected leader set out her priorities outside No 10.

Mrs May was supposed to be the safe pair of hands who could steady the Tory ship after David Cameron's resignation, the Remainer who would ensure Britain left the European Union.

But when she sought a greater mandate in a general election, she lost what majority she had and was forced to seek the support of Democratic Unionist Party MPs.

Mrs May's refusal to be open about the trade-offs between sovereignty and access to EU markets infuriated negotiators in Brussels.

And the withdrawal deal she finally agreed angered Brexiteer ministers and was repeatedly defeated in parliament.

So, amid cabinet resignations, her authority has drained away.

Mrs May was a conscientious, hardworking prime minister imbued with a sense of public duty.

But she was also seen as a dogmatic politician unfit to carry out the task before her - one more Conservative leader brought down by Europe.