Russian President Vladimir Putin said he was ready for "constructive dialogue" with Britain's new premier, while White House spokesman Josh Earnest said USA officials who had worked with May "found her to be quite effective".
Cameron announced his resignation last month, following the UK' shock referendum result to leave the European Union.
She will also attempt to unite her divided Conservative party and a fractured nation in which many, on the evidence of the vote, feel angry with the political elite and left behind by the forces of globalization.
"From the introduction of same-sex marriage to taking people with low wages out of income tax all together, David Cameron has led a one-nation government", she said. "If you're from an ordinary working class family, life is much harder than many people in Westminster realise".
THERESA May came under immediate pressure Thursday on her first full day as Britain's new prime minister after a series of surprise appointments to her cabinet, including the gaffe-prone Boris Johnson as foreign minister.
Euroskeptic former ministers David Davis and Liam Fox were appointed respectively as Brexit negotiator and minister for global trade, two new posts reflecting changed priorities after the referendum.
Theresa May took over as Britain's new prime minister on Wednesday, promising a "bold, new, positive role" for the country less than three weeks after its seismic vote to leave the EU.
On Tuesday night Netanyahu spoke with Cameron and thanked him for standing by Israel, and for the good relations and strengthening of ties between Israel and Britain during his term in office, a statement from the PMO said.
"It's our task to work very closely with governments of ally countries", Merkel told a news conference in Bishkek, Kyrgyzstan, adding there were many problems in the world that made such close cooperation necessary.
Though she was in the Remain camp in the referendum, like Mr Cameron, Mrs May has said there can be no going back on Brexit.
"UK%27s+Leave+Vote+in+EU+Referendum" Cameron resigned immediately after the outcome of June's referendum on the UK's membership to the EU in which 51.9 % of voters had EU+Referendum" supported Brexit.
There is also speculation that May, Britain's second female prime minister - after Margaret Thatcher - will boost the number of women in top posts.
Welsh political figures welcomed her appointment and paid tribute to her predecessor yesterday afternoon.
In his emotional final address to the country, Mr Cameron said that he believed he was leaving the country "much stronger" and the economy "immeasurably stronger" after his six years in office.
And the outgoing Prime Minister shared one last moment in the limelight outside the famous black door with wife Samantha and children Nancy, 12, Arthur, 10 and Florence, five. "I appreciate the maturity of judgment, the steely resolve, the sense of justice and the personal integrity and warmth you have always shown".
In his final question and answer session in the House of Commons on Wednesday, Cameron echoed a line he had once used to taunt former Labour premier Tony Blair.
"I will miss the roar of the crowd".