Mr Sunak said his predecessor was a "relentless campaigner" who had served the British public with distinction.
He said becoming our second female prime minister after Margaret Thatcher was an "extraordinary achievement" for Mrs May. He added: "And actually beyond that, she just has the most amazing track record of dedicated public service over two decades.
"And that is an extraordinary and inspiring example to anyone.
"Anyone who spent time with her knows how committed she is to public service, not just in her community, but the country. And everyone will miss her sorely."
Lord Cameron, who Mrs May succeeded as PM, described her as the "most dedicated of public servants".
He said: "As well as serving as my home secretary and then as PM, she did much to help modernise the Conservative Party and promote women in public life.
"She has been the most dedicated of public servants. The House of Commons will miss her." Mrs May said she had taken the "difficult decision" to quit the Commons after 27 years representing Maidenhead constituency.
The 67-year-old also pledged her support to Mr Sunak's government and said she believed the Tories can win the next election.
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