Jeremy Corbyn, the embattled leader of the opposition Labour Party, congratulated Mr. Cameron for his support for the legalization of same-sex marriage in 2013, and for his efforts to secure the release of Shaker Aamer, a Saudi citizen and British resident, last fall.
She will make appointing a minister to take charge of Brexit one of her first tasks as she enters No 10 as Britain's second female prime minister.
She said that Britain will forge "a bold new positive role" in the world. "Following the referendum, we face a time of great national change. And I know, because we're Great Britain, that we will rise to the challenge".
Cameron, who led the "Remain" campaign, announced the morning after the June 23 referendum that he would stand down, triggering a leadership contest in the ruling Conservative Party.
She promised to fight injustice and inequality with her premiership.
And he said Mrs May would provide "strong and stable leadership in delivering the Conservative manifesto on which we were elected" and wished her well in negotiating "the best possible terms for Britain's exit from the European Union".
Listing his engagements for the day, Mr Cameron joked that "other than one meeting this afternoon with Her Majesty the Queen, my diary for the rest of the day is remarkably light".
Former Mayor of London and leader of the Brexit campaign, Boris Johnson has been appointed foreign minister and Amber Rudd as interior minister.
"We are living through an important moment in our country's history following the referendum", May, the former home secretary, said on Wednesday. The Sun said Cameron had been "undone by his Olympian overconfidence", while the Guardian called him a "prime minister of broken promises".
Earlier, at Prime Minister's Questions - a weekly session, Cameron said May was "a brilliant negotiator". "It's not been an easy journey, and of course we have not got every decision right, but I do believe that today our country is much stronger".
She is to give her first speech as prime minister at 10 Downing Street, and then go alone into her new office and write "The letter of last resort" that informs British nuclear submarines how to operate if the country is threatened with destruction.
May, who supported the campaign to remain in the European Union, said "Brexit is Brexit" and the United Kingdom will leave the bloc with no attempt to "remain through the back door". As May spoke in front of her new residence, a small band of pro-Brexit demonstrators down the street chanted "Theresa May, don't delay!"
She won the Conservative leadership election after her final rival dropped out, and has gone to the palace to formally assume the office of PM.
"So here's an epitaph for Mr Cameron - A dog person who pretended to like cats", he had gone on to say.
Sturgeon said she had a mandate to "to respect the wishes of the people of Scotland to find a way of keeping Scotland within the European Union or protecting our relationship with EU".
Because, yes, we can be pretty tough and test and challenge our leaders - perhaps more than some other countries - but that is something we should be proud of and we should keep at it, and I hope you will all keep at it, and I will will you on as you do. "I will miss the roar of the crowd, I will miss the barbs from the opposition, but I will be willing you on", he said.