Jeremy Corbyn has said Theresa May "underestimated" voters and the Labour Party after the Tories failed to win an overall majority in the election.
He said people had voted "for hope" after his party secured 262 seats in Parliament.
The Labour leader called on Mrs May to resign after the Conservatives were left eight seats short of a majority.
However, Mrs May has said she would form a government with the support of the Democratic Unionists.
In a video post on Twitter, Mr Corbyn thanked those who voted for Labour.
"Your vote for us was a vote for change, a vote for our country and a vote for hope," he said.
He went on to say Theresa May called the general election "in her party's interests, not in the interests of the country" and thought she could "take your vote for granted".
"But she underestimated the Labour Party, and more importantly, she underestimated you."
After Kensington declared for Labour, it now has 262 seats behind the Conservatives' 318, with the SNP on 35 and Lib Dems on 12.
Len McCluskey, leader of the UK's biggest union Unite, said Labour could put together a minority government as Mrs May "can't go on for another five years".
"So my message is that Labour now has to prepare for government," he added.
"And they do that by the Parliamentary Labour Party coming back together and being united and making certain that the policies are developed and spread out and propagated in a way that will be attract more and more people to us."
Earlier, shadow chancellor John McDonnell also suggested Labour could form a minority government.
He told BBC News: "We have laid the foundations for a minority government, and then eventually a majority government."
He said a Labour government would be formed "not through deals or coalitions but policy by policy".
Former Labour cabinet minister Peter Mandelson described the election result was a political "earthquake".
He acknowledged Mr Corbyn, whose leadership he had previously criticised, had been "very sure footed" in the campaign.
But he said if Labour was to come first and not "a good second" next time, the leader had to be much more "ecumenical" in his approach.
Labour took seats from the Conservatives, including Battersea and Canterbury, and unseated former Lib Dem leader and Deputy Prime Minister Nick Clegg in Sheffield Hallam.
The party also made gains in Scotland, which included regaining ex-PM Gordon Brown's former seat of Kirkcaldy and Cowdenbeath.
Speaking following his re-election as the MP for Islington North, Mr Corbyn said: "The prime minister called this election because she wanted a mandate.
"Well the mandate she's got is lost Conservative seats, lost votes, lost support and lost confidence.
"I would have thought that's enough to go, actually, and make way for a government that will be truly representative of all of the people of this country."
Shadow foreign secretary Emily Thornberry described Labour's election result as a "extraordinary performance" which "shows what we can do when we unite".
SNP leader Nicola Sturgeon spoke during the election campaign of a desire to form a "progressive alternative" to the Tories and supporting Labour on a "case by case" basis if the numbers added up. However, Mr Corbyn ruled out a formal deal with the SNP.
Responding to the exit poll, Green Party co-leader Caroline Lucas said: "Any Green MPs elected tonight will do all they can to keep the Tories from Number 10, and back a Labour-led government on a case by case basis."
The Liberal Democrat press office tweeted: "We are getting a lot of calls so just to be clear: No coalition. No deals."
In the 2015 general election, Labour won 232 seats under former leader Ed Miliband, down from the 258 seats secured in 2010.
It went into the 2017 election with 229 seats.
When Prime Minister Theresa May announced the election in April, her Conservative Party had a big double-digit lead in many polls and hoped for a landslide victory. But the campaign saw the Tories' poll lead narrowing.
Labour deputy leader Tom Watson called Mrs May "a damaged prime minister whose reputation may never recover".
Mr Watson said he thought Mr Corbyn's leadership of Labour was safe. "It would be very foolish for anyone to want to stand down in the Labour Party tonight after this result."
But he added: "I can see Boris Johnson sharpening the knives for Theresa May after this result."
The leader of the Unite Union, Len McCluskey, said the election result was "an incredible advance" and it wouldn't be long before there was a Labour government.
He said that Labour's performance reflected that "there is a desperate wish out there for an alternative".
"We've had austerity rammed down our throats, it's brought about inequality, it's brought about the type of society that nobody wants - a low pay, race to the bottom society. Jeremy Corbyn came along and offered an alternative."
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