He died after suffering complications resulting from a recent severe ear infection.
A mathematician of international standing, when Jones accepted the medal in front of the world's leading mathematicians at the International Congress of Mathematicians in 1990, he wore an All Blacks rugby jersey.
"He was a very proud Kiwi," said friend and long-time colleague, Distinguished Professor Marston Conder, from the University of Auckland's Department of Mathematics.
"He was very down to earth, someone you could have a relaxed chat with over a drink, with a great sense of humour, and who did a lot for New Zealand mathematics.
"For many years he wasn't just the only New Zealander to receive the Fields Medal, he was the only Fields Medallist in the Southern Hemisphere.
"This was a huge achievement, awarded only to brilliant mathematicians under the age of 40."
Jones spent his career in the United States but gave generously of his time to the university and to New Zealand mathematics more widely, offering courses and lectures each summer to encourage and mentor younger mathematicians.
He co-founded and led the NZ Mathematics Research Institute (NZMRI) to promote and foster high quality mathematics.
The Royal Society's Te Apārangi Jones Medal, awarded for outstanding achievement in the mathematical sciences, is named after him.
Born in Gisborne, Jones was educated at St Peter's College, Cambridge, before he won a Gillies Scholarship to study at Auckland Grammar.
After graduating from the University of Auckland, he was awarded a Swiss Government Scholarship and completed his doctorate at the University of Geneva. His PhD work there won the Vacheron Constantin Prize.
In the 1980s he worked at the University of California, Los Angeles, and the University of Pennsylvania, before being appointed Professor of Mathematics at the University of California, Berkeley.
From 2011 he held the position of Stevenson Distinguished Professor of Mathematics at Vanderbilt University but remained Professor Emeritus at Berkeley.
His most celebrated work was on knot polynomials, the study of everyday knots of the type used in sailing.
His discovery of what is now called the Jones polynomial had origins in the theory of von Neumann algebras.
"That work in theoretical mathematics, which basically helps distinguish one knot from another, had application in biological science, enabling scientists to identify whether two different types of RNA are from the same source," Conder said.
"That was an outstanding achievement because from theoretical mathematics, his discoveries had real significance in an entirely different field."
Jones described himself in a 2016 interview as "not the model student" and said he "didn't get 'A's' in everything".
And because his fellow students were exceptionally bright, and were awarded most of the available scholarships, it looked like his career might be over before it had begun.
But the Swiss Government scholarship saved the day and his career "blossomed from there".
A keen kite surfer and golfer, he was also a talented choral singer but rarely had time to devote to it.
In 2002 Jones was made a Distinguished Companion of the NZ Order of Merit (DCNZM) (later re-designated Knight Companion KNZM).
He is survived by his wife and three children.
