During the emotional PMQs, Mrs May said: “As a party leader who has accepted when their time is up, perhaps now it’s time for him to do the same?”
The outgoing Prime Minister also blasted Mr Corbyn for not backing her Brexit deal and walking away from talks saying he should be “ashamed of himself”.
Mr Corbyn demanded Boris call a general election as soon as he he is in office so people can “decide their own future”.
The Labour leader even suggested Mrs May could help him to “oppose the reckless plans” of Boris claiming he has “no mandate at all”.
Asked if she was sad to be handing over to Boris, Mrs May said: "I am pleased to hand over to an incoming leader of the Conservative party and Prime Minister who I worked with when he was in my Cabinet and who is committed to delivering on the vote of the British people in 2016 and to delivering for the people of this country."
Mr Johnson will announce his new cabinet today which is expected to include Brexiteers and to increase the representation of ethnic minorities.
Various MPs used the PMQs to praise May May and her commitment to her role, as well as her achievements on issues such as modern slavery, mental health and getting more women into politics.
In her final remarks at the despatch box, Mrs May said she believes there will be another woman prime minister among the current MPs.
She said: “I’m sure that amongst the women in this House today there is a future prime minister, maybe more than one.
“Later today I will return to the backbenches and it will be my first time in 21 years so it’s going to be quite a change from standing here at the despatch box.
Mrs May’s chief of staff Gavin Barwell has said Mrs May did see the prospect of a no-deal Brexit as a threat to the UK staying together.
Mrs May will head to Buckingham Palace this afternoon to formally resign to the Queen.
Mrs May’s final appearance at PMQs was watching by her husband Philip May from the gallery.
Read More on Yahoo News
As Boris enters Number 10, he is expected to address the UK with his vision for the future and life post-Brexit.
There are reportedly “ongoing” conversations between Mr Hunt and Mr Johnson about what the Foreign Secretary’s next role will be.
Mrs May said she will continue her duties from the backbench and her Maidenhead constituents are her priority.