Sir Jeremy passed away on Sunday morning, his wife Suzanne said in a statement.
She paid tribute to a “wonderful father” who “crammed a huge amount into his 56 years”.
She said: “He saw it as a huge privilege to work so closely with four prime ministers and two chancellors and was unwavering in his efforts to help each of them reach their goals.
“He was always conscious of the need for civil servants to see the world through ministers’ eyes while at the same time respecting the boundaries between politicians and civil servants.
“Away from his work, he inspired admiration, respect and affection in his many and diverse group of friends and returned it to them.
“Jeremy could light up any room or conversation and loved hosting a good party.”
In a statement, Prime Minister Theresa May said: “This is extremely sad news and all of my thoughts are with Jeremy’s family and friends.
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“The many retirement tributes paid to Jeremy from across the political spectrum in recent weeks demonstrated his extraordinary talent supporting and advising prime ministers and ministers, and leading the Civil Service with distinction.
“He worked tirelessly to serve our country in the finest traditions of the Civil Service and he is a huge loss to British public life.
“I will always be grateful for the support which he gave me personally and will remember his achievements across his career as we regret that he did not have the chance to offer his talents for longer in retirement.
“Jeremy will be sorely missed and I send my deepest condolences to Suzanne and the children and to all his family and many friends.”
Former Treasury minister Yvette Cooper paid a tearful tribute to Sir Jeremy Heywood as the news of his death broke on Sky News’ Ridge show.
She said: “People will not be aware of quite how many remarkable things he did to steady crises, deal with some of the most difficult problems and as a very honourable public servant to hold people together and to make sure the Government frankly did not do stupid things on many occasions.
“His contribution to public life for all governments, whatever your politics, was immense and we owe him a debt of gratitude.”
Sir Mark Sedwill, Sir Jeremy’s successor as Cabinet Secretary and head of the Civil Service, said he had made “an immense contribution to public life”, serving four prime ministers with distinction “through some of the most challenging episodes of the last 30 years”.
He added: “We will miss him more than we can say, and will be the poorer without his advice, leadership and extraordinary insight.
“He set the highest standards and challenged us to meet them. Jeremy was always looking to move difficult problems forward, restlessly confident to deliver a better way.”