The new leader, who had lent Cameron her formal support in the referendum campaign but is viewed as sympathetic to the Euroskeptic camp, had already said the lineup of ministers would include figures from both sides of the European Union question.
She is under pressure from pro-Brexit Conservatives and other European Union leaders to start formal exit talks with the bloc.
"I'm delighted that for the second time in the British history the new Prime Minister will be a woman and once again a conservative".
She is also expected to meet Scottish First Minister Nicola Sturgeon in the coming days, with Sturgeon telling a press conference Wednesday that Brexit "does not apply to Scotland, as Scotland voted differently in the referendum".
Mrs May arrived at Downing Street from Buckingham Palace, where she officially accepted the Queen's invitation to form a new government.
Cameron then declared his readiness to depart 10 Downing Street, the prime minister's residence, in a scant 48 hours.
Giving up one of his allotted six questions to pay tribute to Cameron, Corbyn asked him to thank his mother for her fashion advice - a nod to a previous rowdy exchange where Cameron said his mother would want the often unkempt Corbyn to "put on a proper suit, do up your tie and sing the national anthem".
"It has been the greatest honor of my life to serve our country as prime minister over these last six years, and to serve as leader of my party for nearly eleven years", he said.
Flanked by his wife and children, the PM said Britain was "much stronger" than when he took over.
Theresa May appointed Britain's new prime minister
"It has been the greatest honor of my life to serve our country as prime minister over these last six years, and to serve as leader of my party for nearly 11 years", he said, accompanied by his wife Samantha and his children - 12-year-old Nancy, 10-year-old Elwen and 5-year-old Florence.
He attends his final Prime Minister's Questions at Parliament on Wednesday, before personally handing his resignation to Queen Elizabeth II. Cameron had campaigned to remain in the bloc.
The Daily Telegraph said in an editorial that Cameron "leaves behind a stronger country, a thriving economy and more chances to get on in life".
Mr. Cameron made a decision to step down after six years as premier following the June 23 referendum on Britain's membership of the European Union, in which 17.4 million people voted to leave the bloc.
"I will miss the roar of the crowd, Cameron told Parliament".
May will seek to calm the country, and the financial markets, after upheaval following the unexpected result in the June 23 referendum.
May was greeted with a huge cheer as she entered the House of Commons for prime minister's questions.
Cameron used his final day in office to try to stake a claim on a legacy beyond his status as the prime minister who - by gambling on a referendum - took Britain out of the EU.
Cameron drew praise from one old adversary, with European Commission President Jean-Claude Juncker saying he would miss the British leader.