"At this time more than anything else, this country needs a period of stability," Mrs May said early on Friday.
"And if, as the indications have shown and if this is correct that the Conservative Party has won the most seats and probably the most votes, then it will be incumbent on us to ensure we have that period of stability - and that is exactly what we will do."
Mrs May - who had a small majority in the previous parliament - had called an early election to try to improve her negotiation positions on Brexit.
But EU Budget Commissioner Günther Oettinger told German radio he was unsure Brexit talks could start later this month as scheduled. He said discussions with a weak UK negotiating partner could lead to a poor outcome.
Mr Corbyn earlier said: "If there is a message from tonight's results, it's this: 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, and make way for a government that will be truly representative of all of the people of this country," he added.
Final election results are expected by Friday lunchtime.
The biggest shock of the night so far has been Liberal Democrat MP Nick Clegg losing his seat to a Labour candidate. He was deputy prime minister of the UK from 2010 to 2015 in a coalition government with the Conservatives.
Former Scottish National Party leader Alex Salmond was also defeated, losing his seat to a Conservative.
A total of 650 Westminster MPs are being elected, with about 45.8 million people entitled to vote. A party needs 326 seats to have an overall majority.
The Conservatives are forecast to win 42% of the vote, Labour 40%, the Lib Dems 7%, UKIP 2% and the Greens 2%.
In the House of Commons, the Conservatives are predicted to be 12 seats short of an overall majority, losing 15. Labour are set to gain about 30, the Lib Dems five and the SNP are predicted to lose 22 seats.
The Green Party would be unchanged with one seat and Plaid Cymru still have three MPs in Wales, according to the poll.
Northern Ireland has different political parties.
How other parties have reacted?
SNP leader Nicola Sturgeon said the election "has been a disaster for Theresa May".
"Her position I think is very, very difficult. We have to wait and see how things shake out," Mrs Sturgeon said, adding that she was "disappointed at the SNP losses".
Leader of UK Independence Party Paul Nuttall tweeted: "If the exit poll is true then Theresa May has put Brexit in jeopardy. I said at the start this election was wrong. Hubris."
Lib Dem President Baroness Brinton said her party could not work with either Labour or the Tories as both are pushing for a "hard Brexit".
Green co-leader Caroline Lucas earlier said she could "hardly dare hope" that the exit poll was right, adding: "To be clear, Greens will never support a Tory government."
Why the election matters - it's all about Brexit
The election will largely determine the UK's negotiation policies in upcoming negotiations with the EU on Brexit.
Theresa May was against Brexit before last year's referendum - but now says there can be no turning back and that "Brexit means Brexit".
The reason the prime minister gave for calling the election was to strengthen her hand during the negotiations.
The Conservatives' priorities were set out in a 12-point plan published in January and the letter formally invoking Brexit in March.
The key elements include:
No longer being bound by EU law and European Court of Justice rulings
Quitting the EU single market and seeking a "comprehensive" free trade deal in its place
Striking trade deals with other countries around the world
The Labour Party campaigned against Brexit in the referendum but now says the result must be honoured, and is aiming for a "close new relationship with the EU" with workers' rights protected.
The party has set out several demands and tests it says Brexit must meet. These include:
Aiming for "tariff-free access" to the EU single market, while accepting "unchanged access" is impossible
Leaving the option of the customs union on the table
When the election exit poll was revealed, the pound immediately dropped by 2% as investors took a position that a hung parliament was a possible outcome, writes BBC's Economics Editor Kamal Ahmed.
Why would that lead the currency to decline? Because a hung parliament means that the government's direction of travel would be less certain.
Deals would have to be done. And those vital Brexit negotiations could become all the more difficult.
Nervousness in the markets is likely to increase and investors could decide to move their money to more attractive places, such as the Eurozone where growth has picked up and political risk has reduced, our editor says.