Zanardi, from Bologna, made his F1 debut in 1991 and later achieved success in the CART series in the United States, winning back-to-back championships in 1997 and 1998.
His life took a dramatic turn in September 2001 when he was involved in a high-speed crash during a CART race in Germany that led to the amputation of both legs.
Zanardi refused to end his sporting career and instead turned to para-cycling, becoming one of Italy's most successful Paralympic athletes.
He won four gold medals and two silver medals at the 2012 London and 2016 Rio Games.
"It is with deep sorrow that the family announces the passing of Alessandro Zanardi, which occurred suddenly yesterday evening, 1 May," his family said in a statement.
"Alex passed away peacefully, surrounded by the love of his family and friends.
"The family would like to express their heartfelt thanks to all those who are showing their support at this time and asks that their grief and privacy be respected during this period of mourning."
Advocate for athletes with disabilities
Zanardi also claimed 12 World Championship gold medals in para-cycling and became a powerful advocate for athletes with disabilities and a source of inspiration for many.
His life suffered another blow in 2020 when he was seriously injured after being struck by a truck while competing in a charity para-cycling relay in Tuscany.
He sustained serious head injuries and spent years undergoing treatment.
"Italy loses a great champion and an extraordinary man, capable of turning every trial of life into a lesson in courage, strength, and dignity," Italy's Prime Minister Giorgia Meloni said.
"Alex Zanardi knew how to get back in the game every time, facing even the toughest challenges with determination, clarity, and a strength of spirit that was truly exceptional."
The International Automobile Federation said Zanardi's journey had "made him one of sport's most admired competitors and an enduring symbol of courage and determination."
F1 President and CEO Stefano Domenicali remembered Zanardi for his "extraordinary strength".
"He faced challenges that would have stopped anyone, yet he continued to look forward, always with a smile and a stubborn determination that inspired us all. While his loss is profoundly felt, his legacy remains strong," he added.
The Italian Olympic Committee called for a minute's silence at all sports events in Italy over the weekend in his honour.
"We've lost a great champion and a great man, who was capable of bouncing back multiple times when faced with life's difficulties," CONI president Luciano Buonfiglio said.