"The death of Alistair Darling, a former chancellor of the exchequer and long-serving member of the Labour cabinet, was announced in Edinburgh today," a spokesperson for the family said in a statement.
"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."
Darling served as chancellor under Gordon Brown from 2007 to 2010, having served as Edinburgh MP from 1987 until he stepped down from parliament in 2015.
Writing on X, formerly Twitter, Brown said: "I am deeply saddened by the death of Alistair Darling. I, like many, relied on his wisdom, calmness in a crisis and his humour.
"I send my deepest condolences to his loving wife Maggie and their children Calum and Anna. He will be missed by all who knew him."
Jim Leaviss, M&G's CIO of public fixed income, said: "Alistair Darling is the only politician I have sent fan mail to. I wrote to tell him what a great job he had done during the Global Financial Crisis in 2008, when he nationalised Northern Rock rather than allowing it to go bust with losses for individuals and companies, and thereby causing a run on all other UK banks.
"He also organised the bailout of the Royal Bank of Scotland. Darling understood the lessons of the Great Depression, a period when politicians decided that failing banks and businesses needed to be liquidated rather than supported.
"Of course, moral hazard, the argument that bailing out bad institutions leads to reckless investment in future, is a real phenomenon and some policymakers may have chosen a different course. But Darling knew that news footage of pensioners queueing to get their money out of high street banks would trigger widespread panic, and that the costs of a bailout might be small compared with the damage done by a UK bank run.
"He was a calm-headed politician just when we needed one, and a member of a government that delivered a fantastic run of low volatility growth and inflation in the decade after Blair's 1997 election win."
Labour leader Keir Starmer said: "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.
"I consider myself incredibly fortunate to have benefited from Alistair's counsel and friendship."
Chancellor Jeremy Hunt described Darling as "one of the great chancellors".
"He will be remembered for doing the right thing for the country at a time of extraordinary turmoil. My deepest sympathies to his family," he said.
In a statement on X, former Labour prime minister Tony Blair said: "Alistair Darling was a rarity in politics. I never met anyone who did not 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.
"I remember him with huge affection. He was taken from us far too soon. My deepest condolences to Maggie, to Calum and to Anna."