"I was the future once", Cameron said.
The Prime Minister will then arrive at her new home in Downing Street - freshly vacated by Cameron.
May will be Britain's second female Prime Minister after Margaret Thatcher, who held office from 1979 to 1990.
He declared: "I said we would confront our problems as a country and lead people through hard decisions".
"And as we leave for the last time, my only wish is continued success for this great country that I love so very much".
She said that Britain will forge "a bold new positive role" in the world, outside of the European Union.
He was replying to a question about Scotland, where most of the electorate voted to stay in the European Union in the referendum.
In a rare personal interview marking her leadership bid, she described how she and her husband Philip had come to terms with not having children, saying that "you accept the hand that life deals you".
In a direct message to voters, she said: "I know you are working around the clock, I know you are doing your best and I know that sometimes life can be a struggle". Both members of Parliament unsuccessfully challenged her for Conservative Party leadership, and Johnson, a former London mayor, was one of the leaders of the campaign to leave the EU.
David Cameron signed off as Prime Minister with a standing ovation from Conservative MPs and applause from some of the opposition, "Armenpress" reports, citing The Telegraph.
Philip Hammond, who had been Foreign Secretary, has succeeded George Osborne as Finance Minister, who has resigned. May's former job of home secretary went to Amber Rudd. May is also expected to boost the number of women in her frontline Cabinet team.
May's predecessor Cameron, appearing earlier in Downing Street with his wife Samantha and their three children, delivered his parting remarks to the nation after six years dominated by the Europe question and the aftermath of the global financial crisis.
Cameron welcomed Leadsom's decision to drop out of the race and said he was confident May would steer the country in the right direction, calling her strong and competent, and offering her his full support.
Last but not least, he paid tribute to "the love of my life" Samantha who "has kept me vaguely sane".
Theresa May has become Britain's prime minister after meeting with Queen Elizabeth II.
Thomas de Maiziere has been outgoing Home Secretary May's German counterpart for several years as his country's top security official. She now faces the daunting task of leading the country out of the European Union.
She will reportedly give priority to appointing a Cabinet minister in charge of implementing Brexit, the decision to leave the EU.
The new leader stressed she would fight against social injustice and that she believes in the unity of all aspects of the United Kingdom. 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!"
After months of simmering discontent among Labour politicians with party leader Jeremy Corbyn, the conflict exploded into the open after the referendum when the vast majority of the politicians rejected his leadership.
Theresa May's entrance into office "is one of the few calls for stability in British politics in what has been an extremely tumultuous time", Klaas says.
He said his government had cut the deficit, overseen economic growth and legalized same-sex marriage. "In the end, public service, the national interest, that's what it's all about".
"I will miss the roar of the crowd". "I will miss the barbs from the opposition".
They stood by his side as he gave his final speech as Prime Minister.