"In the United States Senate I can think of no-one who engendered greater respect or affection from members of both sides of the aisle," he said.
"His ideas and ideals are stamped on scores of laws and reflected in millions of lives," he said.
Former President George W Bush, who was criticised by Sen Kennedy over Iraq, described him as "a man of passion who advocated fiercely for his convictions".
House Speaker Nancy Pelosi, a Democrat, praised his determination to make quality health-care available to all Americans, while Senate Minority leader Mitch McConnell, a Republican, also added his voice.
"No one could have known the man without admiring the passion and vigour he poured into a truly momentous life," he said.
The BBC's Richard Lister in Washington said Senator Kennedy, known affectionately as Teddy, would be remembered as one of the most effective legislators in American history.
Daniel Sandford, BBC News, Washington
President Obama was leading the tributes, saying that Senator Kennedy was the greatest US senator of our time.
But the praise was coming from right across the political divide. The Republicans were also praising him, saying that he was the kind of man that you couldn't help but like even if you disagreed with him.
That kind of praise has been echoing across the morning shows - they all broke into special coverage of the kind that is normally reserved for when former presidents die.
But of course all of the coverage has also included the controversies in Edward Kennedy's life, not least of all the death of Mary Jo Kopechne at Chappaquiddick bridge in 1969.
He was skilled at forging alliances across party lines: pushing an education initiative with Mr Bush and immigration reform with Republican John McCain.
But he was a fierce critic of the Bush administration too - particularly on Iraq and when allegations of US military abuses there emerged.
He will also be remembered as a staunch supporter of Irish nationalism - at one time calling for British troops to leave Northern Ireland - although he was later involved in the peace process leading to the Good Friday Agreement, our correspondent adds.
In the UK, Prime Minister Gordon Brown said Sen Kennedy would be mourned in every continent. "I am proud to have counted him as a friend," he said.
And UN Secretary General Ban Ki-moon praised him as a man who "was a voice for those who would otherwise go unheard".
Edward Kennedy was, at his death, the third-longest senator in US history.
EDWARD MOORE KENNEDY
1932 Born, youngest of nine children
1962 Becomes country's youngest senator
1963, 1968 Brothers President John F Kennedy and Senator Robert F Kennedy both assassinated
1969 "Chappaquiddick incident" - Kennedy flees scene after road crash in which his young passenger dies
1980 Runs unsuccessfully for Democratic nomination against sitting President Jimmy Carter
He became a Massachusetts senator in 1962, replacing his brother John when he resigned to become president.
He was the only one of four brothers not to die a violent death. His brother Joseph was killed in an air crash in World War II, and both John F Kennedy and presidential hopeful Robert F Kennedy were assassinated in the 1960s.
He was widely expected to be the next Kennedy in the White House, but he was never able to fully overcome a scandal in 1969 when he drove a car off a bridge at Chappaquiddick near his home, killing his female passenger.
The incident helped derail his only presidential bid, more than a decade later.
But he remained active in politics right up until his death, famously endorsing Barack Obama for the Democratic nomination during a tight race with Hillary Clinton last year.
US MEDIA REACTION TO TED KENNEDY'S DEATH
Kennedy was at the center of the most important issues facing the nation for decades, and he did much to help shape them. A defender of the poor and politically disadvantaged, he set the standard for his party on health care, education, civil rights, campaign-finance reform and labor law
He was a Rabelaisian figure in the Senate and in life, instantly recognizable by his shock of white hair, his florid, oversize face, his booming Boston brogue, his powerful but pained stride. He was a celebrity, sometimes a self-parody, a hearty friend, an implacable foe, a man of large faith and large flaws, a melancholy character who persevered, drank deeply and sang loudly. He was a Kennedy.
Seared in my memory: When I interned at the Heritage Foundation, I would pop into Mass at Saint Joseph's on the Hill. And I would almost always find myself sitting near Ted Kennedy. He's responsible for things that are deeply offensive to my conscience and diametrically opposed to the teachings of the Catholic faith, and he probably led some people astray by his example. But our faith also teaches that we are all sinners and that there is redemption. He had some incredibly good forces in his life, not least among them his sister, Eunice, who just died. I pray for the repose of his soul. R.I.P. Senator Kennedy.
Elected first in 1962, the 77-year-old Massachusetts liberal was rooted in the civil rights and Great Society battles of that decade, but his enduring strength was an ability to renew himself through his mastery of issues and the changing personalities of the Senate. Nowhere was this clearer than in Kennedy's early support of Barack Obama in 2008, when the young Illinois Democrat needed to establish himself against more veteran rivals for the White House. Kennedy not only campaigned for Obama but, at risk to his own health, opened the Democratic National Convention a year ago in Denver and returned to Washington repeatedly last winter to cast needed votes to move the new president's economic recovery agenda.
In many ways, he was the last man standing, straddling a mythic family mantle of fame and a vaunted career of political service, all the while wearing the crown of Camelot decades after its heyday...the senator's death brought to a close a storied political era - of assassinations, Jackie O, Palm Beach, Chappaquiddick - and a lifetime of both tragedy and public service.
In losing Kennedy, Obama loses a key Senate dealmaker at a crucial moment in legislative negotiations over the health care bill. Though an icon of Democratic liberalism, Kennedy was known to colleagues as a jovial pragmatist, whose many friendships with colleagues across the political and ideological spectrum made him one of the Senate's most influential players.
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