He was a member of parliament from 1987 until he stepped down in 2015.
Tributes from all sides of the political divided have been paid to Mr Darling who served as chancellor in Gordon Brown’s government after Tony Blair stood down as Prime Minister.
He served as chancellor from 2007 to 2010 and helped steer the country through the 2008 financial crisis.
A statement issued on behalf of Mr Darling's family said: “The death of Alistair Darling, a former Chancellor of the Exchequer and long-serving member of the Labour cabinet, was announced in Edinburgh today.
“Mr Darling, the much-loved husband of Margaret and beloved father of Calum and Anna, died after a short spell in Western General Hospital under the wonderful care of the cancer team.”
Labour leader Sir Keir Starmer said: “I am deeply saddened to learn of the passing of Alistair Darling. My heart goes out to his family, particularly Maggie, Calum and Anna, whom he loved so dearly.
“Alistair lived a life devoted to public service. He will be remembered as the Chancellor whose calm expertise and honesty helped to guide Britain through the tumult of the global financial crisis.
“He was a lifelong advocate for Scotland and the Scottish people and his greatest professional pride came from representing his constituents in Edinburgh.
“I consider myself incredibly fortunate to have benefited from Alistair’s counsel and friendship. He was always at hand to provide advice built on his decades of experience – always with his trademark wry, good humour.
“Alistair will be missed by all those whose lives he touched. His loss to the Labour Party, his friends and his family is immeasurable.”
Former prime minister Sir Tony Blair said Mr Darling was the “safest of safe hands” in government.
“Alistair Darling was a rarity in politics,” Sir Tony said.
“I never met anyone who didn’t like him. He was highly capable, though modest, understated but never to be underestimated, always kind and dignified even under the intense pressure politics can generate.
“He was the safest of safe hands. I knew he could be given any position in the Cabinet and be depended upon. I liked him and respected him immensely as a colleague and as a friend.
“In all the jobs he did for me in Government – chief secretary, work and pensions, transport, trade and industry and of course as secretary of state for Scotland, he was outstanding.
“He could take tough decisions on spending when he needed to, but as he did with Crossrail, when convinced of a project’s importance, he would be equally tough in supporting it.
“I remember him with huge affection. He has been taken from us far too soon. My deepest condolences to Maggie, to Calum and Anna.”
Tributes also came from current Prime Minister Rishi Sunak who said his death was "a huge loss to us all".
Mr Sunak said: "He was a dedicated public servant who served this country though challenging times.“The role he played during the 2014 independence referendum was vital in keeping our union together.“My deepest condolences go out to his family and friends at this difficult time.”Former prime minister Boris Johnson said Mr Darling “was a towering figure in Labour politics and he always brought wit, wisdom and intellect to his work”.
Former shadow business secretary Chuka Umunna, who worked closely with Alistair Darling, told the Standard that Mr Darling was “a really great man and an excellent steady hand through a very, very difficult time for the country. It’s a huge, huge loss.
"He was always incredibly helpful and generous. He was a thoroughly decent and humble man, there was no ego to him at all.”
Serving chancellor Jeremy Hunt said it was "a sad day".
Writing on X, he described Mr Darling as "one of the great Chancellors" who did "the right thing for the country at a time of extraordinary turmoil".
A sad day – I want to pay particular tribute to one of my predecessors, Alistair Darling. One of the great Chancellors, he’ll be remembered for doing the right thing for the country at a time of extraordinary turmoil. My deepest sympathies to his family.
— Jeremy Hunt (@Jeremy_Hunt) November 30, 2023
Former shadow chancellor John McDonnell wrote on X: "So sorry to hear that Alistair Darling has died.
"We may have at times differed on economic strategy but he was always generous of spirit & I enjoyed his quick wit in even the most challenging of times. I send my condolences and deepest sympathy to his family and friends."