Dundee, who oversaw the career of Muhammad Ali, is reported to have died of a heart attack in Florida having suffered from a blood clot in recent weeks.
Inducted into the Boxing Hall of Fame in 1994, Dundee was among the sport's most admired characters.
He oversaw the careers of Sugar Ray Leonard, Jimmy Ellis and George Foreman, among others, but it is his work with Ali that defines his legacy.
Cornerman to Ali for much of his 21 years as a professional boxer, Dundee was present for all of the three-times world champion's greatest moments.
"Angelo died surrounded by family and friends," his family said in a statement.
"He was very happy that he got to celebrate Ali's 70th birthday earlier this year and also that he got to go to the Hall of Fame.
"He still had much to do, but led a full and extraordinary life."
Dundee's son Jim told the Miami Herald: "He was recuperating and coming along quite well.
"He was already making plans to (go) to Las Vegas for another event in two weeks.
"Thankfully the whole family was with him. We have lost a great man.
"My dad led a wonderful life. Sadly, many of the great people from that generation are gone. This is the end of an era."
Freddie Roach, currently boxing's most respected coach who includes Manny Pacquiao and Amir Khan in his stable, paid tribute to Dundee.
"Angelo is Dundee was one of my idols," Roach told Sky Sports News.
"I always used to say I wanted to be like him because he doesn't talk badly about anybody, but I'm not quite there yet.
"A great man and great loss to our sport. We'll miss him deeply."
Britain's former heavy and cruiserweight world champion David Haye used Twitter to hail Dundee's credentials as boxing's finest trainer.
"R.I.P. Angelo Dundee. The Greatest Coach ever," said Haye.
"I had the pleasure of spending time with him in Miami last year. What a true gentleman."
Dundee will be remembered in England for his role in Ali's defeat of British icon Henry Cooper at Wembley in 1963.
Cooper floored Ali - then still using his birth name of Cassius Clay - at the end of the fourth round in London with a trademark left hook which scrambled the American's senses.
Ali was saved first by the bell, however, and then by Dundee, who breached the rules by helping him to the corner.
Dundee then illegally used smelling salts to help his fighter regain his senses and made a small tear in one of Ali's gloves, forcing a welcome delay while a new pair were sought. Ali then went out and beat Cooper on a cuts stoppage in the very next round.
Dundee, whose wife died in 2010, is survived by his son Jim and a daughter, six grandchildren and a great grandchild.
Angelo Dundee factfile
1921: August 30 - Born Angelo Mirena in Philadelphia.
1955: Helps Carmen Basilio win the world welterweight title against Tony DeMarco.
1960: Begins training one-fight novice Muhammad Ali, then still using his birth name Cassius Clay.
1964: Plays a crucial role in Ali avoiding defeat to Henry Cooper in London, illegally helping his fighter to the corner and using smelling salts after he was knocked down by a left hook. Allegedly tears a hole in one of Ali's gloves to buy more time for his fighter to recover.
1965: Ali defeats Sonny Liston to become world heavyweight champion.
1971: Ali, back from boxing exile, loses for the first time to Joe Frazier at New York's Madison Square Garden.
1974: January 28 - Helps Ali avenge Frazier defeat, winning points decision in rematch.
October 30 - Ali shocks George Foreman in the Rumble in the Jungle, playing 'rope a dope' before stopping the heavy-hitting younger man in the eighth.
1975: Ali beats Frazier in the 'Thrilla in Manila' with an epic 14th-round stoppage victory.
1977: 'Sugar' Ray Leonard turns professional with Dundee in his corner.
1979: Leonard wins WBC welterweight title by beating Wilfred Benitez.
1980: Leonard loses to Roberto Duran.
1981: Leonard beats great rival Thomas Hearns in 'fight of the year', stopping the Detroit man in the 14th round. Dundee famously rallies his charge, warning him "You're blowing it, son!" before Leonard scores the knockout.
Ali retires following ignominious defeat by Trevor Berbick.
1987: Leonard returns to the ring to beat Marvin Hagler for the WBC middleweight belt.
1988: Dundee and Leonard split for the fighter's latest comeback, against Donnie Lalonde.
1992: Inducted into prestigious International Boxing Hall of Fame.
1994: Works Foreman's corner as the veteran knocks out Michael Moorer to regain the world heavyweight title.
2012: January - Attends Ali's 70th birthday celebrations in Louisville, Kentucky.
February 1 - Dies of a heart attack in Florida.