Britain's Labour leader Jeremy Corbyn has said Theresa May was right to resign as prime minister, saying whoever replaced her as leader of the Conservative Party must call an election.
This will trigger a leadership contest in the party with the winner becoming the next British prime minister.
"She has now accepted what the country has known for months: she cannot govern, and nor can her divided and disintegrating party," Mr Corbyn said in a statement.
"The Conservative Party has utterly failed the country over Brexit and is unable to improve people’s lives or deal with their most pressing needs. Parliament is deadlocked and the Conservatives offer no solutions to the other major challenges facing our country.
"Whoever becomes the new Conservative leader must let the people decide our country’s future, through an immediate General Election."
A number of members of Mrs May's cabinet also tweeted messages during and after her brief statement to the media.
International Trade Secretary Liam Fox tweeted that Mrs May "acted with dignity and honour in pursuit of what she believes to be in the national interest, the hallmark of her time in public life."
.@theresa_may has acted with dignity and honour in pursuit of what she believes to be in the national interest, the hallmark of her time in public life.— Dr Liam Fox MP (@LiamFox) May 24, 2019
Work and Pensions Secretary Amber Rudd tweeted: "The Prime Minister has shown great courage. "She is a public servant who did all she could to bring Brexit to a resolution. "Her sense of duty is something everyone should admire and aspire to."
Foreign Secretary Jeremy Hunt tweeted: "I want to pay tribute to the PM today. Delivering Brexit was always going to be a huge task, but one she met every day with courage & resolve.
"NHS will have an extra £20bn thanks to her support, and she leaves the country safer and more secure. A true public servant," he added.
Andrea Leadsom, who resigned as House of Commons Leader on Wednesday, said Mrs May did her utmost in the role.
A very dignified speech by @theresa_may. An illustration of her total commitment to country and duty. She did her utmost, and I wish her all the very best— Andrea Leadsom MP (@andrealeadsom) May 24, 2019
Scotland's First Minister Nicola Sturgeon tweeted: "I wish Theresa May well. She and I had profound disagreements - not least on her handling of Brexit and her disregard for Scotland's interests. However, leadership is tough - especially in these times - and she deserves thanks for her service.
"Her departure will not solve the Brexit mess that the Tories have created. Only putting the matter back to the people can do that. Given current circumstances, it also feels deeply wrong for another Tory to be installed in Number 10 without a General Election."
Taoiseach Leo Varadkar paid tribute to Theresa May following her announcement.
In a statement, Mr Varadkar said that he got to know Mrs May very well over the last two years.
"She is principled, honourable, and deeply passionate about doing her best for her country, and her party," he said.
"Politicians throughout the EU have admired her tenacity, her courage, and her determination during what has been a difficult and challenging time."
Mr Varadkar added that Mrs May "strove to chart a new future for the United Kingdom".
The Taoiseach said he wishes Mrs May well for the future and looks forward to "working closely" with her successor.
Dutch Prime Minister Mark Rutte tweeted "thanks and respect" to Mrs May, adding that the "agreement reached between the EU and the United Kingdom for an ordered Brexit remains on the table".
French President Emmanuel Macron stands ready to work with a new British prime minister on all European and bilateral issues after Mrs May's resignation, but wants clarity on Britain's approach to Brexit, an Elysee official has said.