His family said they were devastated to lose a "fine man and loving father". No cause of death has been given but police said it was not suspicious.
Mr Kennedy, who led his party from 1999 to 2006, lost his seat last month.
David Cameron, Nick Clegg and Nicola Sturgeon have led tributes, while former Prime Minister Tony Blair said the death was an "absolute tragedy".
'Great sadness'
Mr Clegg, who resigned as Liberal Democrat leader last month, said that on a good day Mr Kennedy had had "more political talent in his little finger than the rest of us put together".
"Charles devoted his life to public service, yet he had an unusual gift for speaking about politics with humour and humility which touched people well beyond the world of politics," he said.
"He was one of the most gentle and unflappable politicians I have ever known, yet he was immensely courageous too not least when he spoke for the country against the invasion of Iraq."
Mr Kennedy's family said in a statement: "It is with great sadness, and an enormous sense of shock, that we announce the death of Charles Kennedy.
"We are obviously devastated at the loss. Charles was a fine man, a talented politician, and a loving father to his young son."
A Police Scotland spokesman said: "Police officers attended an address at Fort William on Monday, June 1 to reports of the sudden death of a 55-year-old man. Police were notified by ambulance service personnel. There are no suspicious circumstances."
'A different style of politics'
By Norman Smith, BBC assistant political editor
Charles Kennedy was one of the most influential politicians of his generation. In 2005 he led the Liberal Democrats to their best election result, carved out a distinctive position for his party on the left of British politics and, perhaps most significantly, ensured his party was at the forefront of opposition to the Iraq War.
Charles Kennedy also brought a different style of politics to Westminster; more informal; relaxed and outgoing; generous to opponents, warm to friends and not one for the more bitter, dark arts of politics.
He was a politician as much at ease in the television studios as in the Chamber and struck a chord with the public in an age when politicians were more reserved and removed. He sought to fashion a different way of doing politics.
Never the most organised of politicians, he found the business of leadership more onerous and that, coupled with his drinking problems, fuelled disquiet within the Parliamentary party that was eventually to lead to his toppling.
Mr Kennedy's political career began in the Social Democratic Party and he became the youngest MP of the time at the age of 23 when he won the Ross, Cromarty and Skye seat in 1983.
At first he was SDP spokesman on social security, Scotland and health and when most of his party merged with the Liberals to form the Lib Dems in 1988, he continued to hold a series of frontbench posts.
He took over the Liberal Democrat leadership from Paddy Ashdown in 1999 and led the party to its best election result since the 1920s in 2005, when the Lib Dems won 62 seats.
In January 2006 he said he had been receiving treatment for an alcohol problem and resigned as leader.
'Chatshow Charlie'
After this, Mr Kennedy maintained a lower profile. He did not play any role in the coalition government, having voted against his party's decision to enter an alliance with the Conservatives.
Writing in May 2010, he said the tie-up with the Conservatives drove "a strategic coach and horses through the long-nurtured 'realignment of the centre-left' to which leaders in the Liberal tradition, this one included, have all subscribed" since the 1950s.
During the 1990s, Mr Kennedy built his profile through TV appearances, earning him the nickname, which he hated, of "Chatshow Charlie".
His 2002 marriage to Camelot public relations executive Sarah Gurling was seen by many in the party as a sign he was settling down.
The birth of his son in 2005 was seen as a further sign that the hard-partying Kennedy - one commentator had dubbed him "Jock the lad" - was being transformed into a family man.
'We became chums'
By Brian Taylor, BBC political editor, Scotland
I remember in 1983 when he was first elected. He was 23 years old. It was a rather unexpected victory.
He had been on a scholarship in the US and returned, and after a couple of weeks campaigning found himself an MP after taking a Conservative seat in Ross and Cromarty.
I was in the Westminster lobby at the time as a slightly older lobby correspondent. We got together and became chums. I was working for the Press and Journal. He was a very senior MP for me to be covering.
He was a remarkable individual even then. Talented and driven, he always struck me though as being slightly remote from partisan politics. He never had that killer instinct. He could always see the other side of the perspective.
He was exceptionally gifted.
Tributes to Mr Kennedy have poured in from across the political spectrum.
Prime Minister David Cameron said he was "deeply saddened" by the news. He tweeted: "He was a talented politician who has died too young. My thoughts are with his family."
Acting Labour leader Harriet Harman said his death was a huge loss, describing Mr Kennedy as "a delightful person with a great intellect".
'Great company'
Tony Blair, who clashed with Mr Kennedy over the former Labour prime minister's decision to take the UK to war in Iraq, said he had the greatest of respect for a man who had entered Parliament at the same time as him, in 1983.
"Charles's death is an absolute tragedy. He was throughout his time a lovely, genuine and deeply committed public servant," he said.
"As leader of the Liberal Democrats, we worked closely together, and he was always great company, with a lively and inventive mind."
Scotland's First Minister Nicola Sturgeon tweeted: "Sad beyond words to hear the news about Charlie Kennedy. A lovely man and one of the most talented politicians of his time. Gone too soon."
Lord Ashdown told the BBC that Mr Kennedy "had his demons, we all have our demons, but on form and when he was on song Charles was the best of all of us".
"It has been a very difficult time for him, he lost his seat, at the beginning of the election campaign he lost his father.
"He was, I think, a person who maybe characterised a different and more welcome political age when politicians generally spoke the language of ordinary people and stuck to the things they believed in… which is why he was so popular."
And former Lib Dem MP Vince Cable paid tribute to Mr Kennedy's "shrewd political judgement" and "political courage under attack".
"He gave leadership not just to the party but to the country in opposition to the Iraq war," he said.
