The United States congratulated May and said it was confident in her ability to steer Britain through the Brexit negotiations.
May, 59, is the second woman Prime Minister in British history after Margaret Thatcher's tenure from 1979 to 1990.
Cameron - stepping down after losing the June Brexit referendum - praised May as a "brilliant negotiator".
"Of course Theresa is going to want to make sure she's got a balanced ticket that represents the views of different parts of the party", Cabinet minister Chris Grayling, who campaigned for Brexit and managed May's leadership campaign, told BBC radio. Johnson led the campaign to drop out of the EU.
The same company that moved the Camerons into Downing Street in 2010 arrived Tuesday to take them out again.
Cameron said it was the "greatest honour" of his life to serve as prime minister.
Cameron said that he was sure that May would provide "strong and stable leadership" and wished her luck during the European Union talks as he delivered his farewell speech. Since then, the queen has given her royal seal of approval to a dozen prime ministers atop 16 governments.
Earlier Wednesday, Cameron took questions from members of Parliament at the House of Commons, which ended in a standing ovation from lawmakers.
May's ascension to Downing Street was received by numerous well wishes from leaders around the world - including Jean-Claude Juncker, the executive leader of the European Union, who offered his "warmest congratulations" and an invitation to discuss Britain's exit in the very near future.
But as she looks to bridge the divide between those wanting to leave the European Union and those who wished to remain, May is also set to face her first Prime Ministerial discussions this evening with her big-hitting global counterparts.
Former energy secretary Amber Rudd will take Prime Minister Theresa May's old post of interior minister.
"That will be the mission of the government I lead, and together we will build a better Britain".
He attends his final Prime Minister's Questions at Parliament on Wednesday, before personally handing his resignation to Queen Elizabeth II.
An early advocate of Conservative "modernisation" in the wilderness years that followed, Mrs May quickly joined the shadow cabinet in 1999 under William Hague as shadow education secretary and in 2002 she became the party's first female chairman under Iain Duncan Smith.
The foreign secretary position is one of the most high profile in the British government and Johnson's appointment comes as a surprise because he was widely acknowledged to misstep in the immediate aftermath of the U.K.'s vote to leave the European Union. Colleagues have described her as tough-minded and well-briefed on her portfolio of issues at home affairs, which included the fight against Islamist extremist violence and policing of the country's borders.
Trying to reclaim his legacy from Brexit, Cameron said his government had cut the deficit, overseen economic growth and legalized same-sex marriage.
Although she favoured Britain remaining in Europe, May has repeatedly declared that "Brexit means Brexit" and that there can be no attempt to reverse the referendum outcome.
Gesturing across the room and up to his wife, Samantha, watching from the gallery, he said: "I will miss the roar of the crowd, I will miss the barbs from the opposition, but I will be willing you on".
Reflecting on the power of politicians to drive through change, Mr Cameron went on: "The last thing I'd say is that you can achieve a lot of things in politics, you can get a lot of things done".