The new US President met Gordon Brown last July at Downing Street
Gordon Brown has heralded the inauguration of President Barack Obama as a "new chapter in both American history and the world's history".
The prime minister praised Mr Obama as a "man of great vision and moral purpose" who would tackle the global financial crisis and other problems.
Conservative leader David Cameron called Mr Obama's inauguration a "hugely momentous occasion".
Lib Dem deputy leader Vince Cable said Mr Obama was an "inspiration".
Huge crowds packed Washington's National Mall to watch the new president take the oath of office, taking over from George W Bush.
Speaking in Downing Street, Mr Brown said: "The whole world is watching the inauguration of President Obama, witnessing a new chapter in both American history and the world's history.
"He's not only the first black American president but he sets out with the determination to solve the world's problems."
Barack Obama has said people need to work together to achieve change and I agree with him
David Cameron, Conservatives
Mr Brown said: "He is a man of great vision, he is a man of great determination and energy, and he is a man of great moral purpose."
The prime minister added that Mr Obama was clear that these were "extraordinary times" with "unprecedented challenges".
The new president had already shown "great courage" and "boldness" in setting out plans to combat the financial crisis.
Earlier on Tuesday Mr Brown told cabinet ministers he aimed to work closely with the new president on issues including the Middle East peace process, Iraq, Afghanistan and the economy.
He also said Mr Obama's administration shared "a very similar view on the role of the environment".
Mr Brown met Mr Obama last July at Downing Street during the then presidential candidate's European tour. Mr Obama also met former prime minister Tony Blair and Mr Cameron on the trip.
British politicians have argued over what his election means.
Mr Cameron has suggested it means voters wanted change, while Mr Brown has said Mr Obama represents "progressive" values shared by Labour.
Speaking at a party organised by the campaign group Operation Black Vote to celebrate Mr Obama's inauguration, Mr Cameron said: "It's a hugely momentous occasion.
"I think a lot will change. Barack Obama has said people need to work together to achieve change and I agree with him."
Mr Cable said: "This man is an inspiration, with many expectations upon him.
"He is asked to solve an economic crisis, a war in the Middle East, to heal divisions in America that have lasted generations. He is also asked to solve even more complicated problems, such as global warming.
"There will be people questioning his achievements but we are getting someone with a new, positive approach to seemingly intractable problems."