'In the next couple of months we may see the election of a Eurosceptic prime minister who wants to repudiate the Withdrawal Agreement and go for a no-deal,' he said.
Austrian chancellor Sebastian Kurz praised Theresa May as a 'principled' leader and voiced the hope that her successor will 'see to an orderly Brexit'
'Or we may even see a new British Government that wants a close relationship with the EU and goes for a second referendum.
'Whatever happens we are going to hold our nerve, we are going to continue to build and strengthen and deepen our alliance across the European Union, and we will make sure we see Ireland through this.'
His foreign minister Simon Coveney said the EU was unlikely to offer a better deal to Mrs May's successor.
'From my perspective, I don't see the European Union offering any new prime minister a better or very different deal to what was on offer to Theresa May,' 'This idea that a new prime minister will be a tougher negotiator and will put it up to the EU and get a much better deal for Britain? That's not how the EU works.'
Irish premier Leo Varadkar said Mrs May's resignation could signal a 'very dangerous' phase for his country, he was 'sorry to hear' of Theresa May's resignation and thanked her for her efforts on Brexit
A European Commission spokeswoman said its president Jean-Claude Juncker had 'followed Prime Minister May's announcement this morning without personal joy'.
'The president very much liked and appreciated working with Prime Minister May, and has said before Theresa May is a woman of courage for whom he has great respect,' the spokeswoman said.
'He will equally respect and establish working relations with any new prime minister, whomever they may be, without stopping his conversations with Prime Minister May.
'Our position on the Withdrawal Agreement has been set out by my colleague yesterday. There is no change to that. 'The European Commission and the Article 50 format has set out its position and we remain available for anyone who will be the new prime minister.'
EU negotiator Michel Barnier offered his 'full respect' to the outgoing PM, thanking her for her 'determination in working towards the UK's orderly withdrawal'.
'What could happen now, let me just clearly say here in Brussels that it is for the UK to decide. Nobody else,' he said.
EU Brexit negotiator Michel Barnier (pictured) said it was 'for the UK to decide' what happens next after Theresa May's resignation
Many world newspapers led with images of Theresa May breaking down with emotion as she quit. Here, the website of German tabloid Bild simply declares 'Theresa May: Resignation' and quotes her regret that she could not get Brexit over the line
There was little warmth from Russia, where Vladimir Putin's spokesman said Mrs May's tenure had been a 'difficult period in our bilateral relations'.
Anglo-Russian relations fell to their lowest ebb in decades after former spy Sergei Skripal was poisoned in Salisbury last year.
Dutch leader Mark Rutte said he had given Mrs May his 'thanks and respect' and said the withdrawal deal she negotiated remained on the table.
In Austria, chancellor Sebastian Kurz - who is facing a wave of domestic pressure himself - said Mrs May was a 'principled and head-strong politician'.
'I wish her well. I hope that despite her announcement, reason will prevail in the UK and her successor will see to an orderly Brexit,' he said.
Belgian Prime Minister Charles Michel said: 'In spite of our differences Theresa May showed an openness of mind to resolve Brexit for UK and EU. I want to thank her for her resolve and cooperation.'
German broadsheet newspaper Die Welt said that Mrs May had resigned 'with a cracking voice', explaining that she had failed to get her Brexit deal through Parliament
Czech leader Andrej Babis said he hoped Britain might hold a second referendum and vote to remain in the EU after all French broadcaster TV5 Monde said that the 'unbreakable woman has just given way' after Mrs May withstood months of pressure before finally resigning
There was no immediate response from the White House where it was very early in the morning when Mrs May made her Downing Street statement.
Democratic congressman Brendan Boyle said he wished Mrs May well and said it was 'imperative that her successor does nothing to risk the Good Friday Agreement' in Northern Ireland.
A rare piece of high praise came from Gibraltar, where the British territory's chief minister Fabian Picardo said: 'Thank you Prime Minister for a job well done against impossible odds'.
When many others would have been cowed by the impossible odds, the daunting headlines and the impenetrable politics of Brexit, she has ploughed on without complaint,' he said.
'And when it came to Gibraltar, she was steadfast in our defence and rock solid in her resolve that we would be included in any Withdrawal Agreement that she might finally propose.'
'History will judge you more warmly and positively than today's fickle headlines.'
The New York Times said Theresa May had been 'undone by Brexit' as her tumultuous three-year premiership comes to a close
Meanwhile Indian leader Narendra Modi sent a curiously-timed tweet, promising to work with Britain after his own re-election, only shortly after Mrs May had resigned.
Many foreign news websites led with the striking image of Mrs May breaking down in tears as her premiership finally crumbled.
German national newspaper Die Welt ran with the headline: 'With a cracking voice, May announces her resignation'.
The New York Times said Britain's PM had been 'undone by Brexit', predicting a 'vicious contest to succeed her within the Conservative party'.
France's TV5 Monde had a more poetic headline, saying: 'The unbreakable woman has just given way'.
Der Spiegel in Germany said she had been 'overcome by emotion', saying she had been 'massively under pressure from all sides'.
Chapter Two
Childhood and Early Life and Parents
She was born Theresa Mary Brasier on October 1, 1956, into a middle-class family in Eastbourne, Sussex. She was the daughter of Church of England vicar, Hubert, and his wife, Zuaidee.
Her father was a Church of England clergyman who held the role of Vicar of Wheatley, and her grandfather had been a regimental sergeant major.
Her father died from injuries sustained in a car crash when she was only 25. She was raised in Oxfordshire, and attended a state primary school, an independent Covent school, and a grammar school in the village of Wheatley.
As a young girl she attended several schools including Heythrop Primary School, Oxfordshire, St. Juliana's Convent School for Girls, and the Wheatley Park Comprehensive School.
She then proceeded to the University of Oxford where she studied geography at St Hugh's