Former world No1 player Ballesteros had been diagnosed with a brain tumour in 2008.
Bernard Gallacher, who captained Ballesteros in three Ryder Cups between 1991 and 1995, was among the first to react, saying: “America had Jack Nicklaus and Arnold Palmer — Seve was our Arnold Palmer and Jack Nicklaus rolled into one.
“You can't speak too highly of him, Seve was Europe's best ever player.”
The flags at Real Club de Golf El Prat in Barcelona flew at half-mast in his memory for today's third round of the Spanish Open, while the players wore black ribbons. They also observed a minute's silence.
Julian Small, chief executive of Wentworth Golf Club, where Ballesteros won five titles, said: “Seve was a great champion. He was a swashbuckling man, a handsome man who really entertained the crowd.
“He played such a fantastic game of golf. Wherever Seve went there was always a huge crowd. He saw the West course here at Wentworth as one of his favourite places to play.
“He is the pioneer, he is the person that has changed the shape in many ways — especially with the general public — of European golf. His way of playing, his whole charisma, he brought it to new markets, to new people, to new populations. He is Europe's version of Arnold Palmer, he is the person that really made that big difference.
“It's legendary what he did. To think that Colin Montgomery had him on a speakerphone to the Ryder Cup team last year down is a sign of how the professional golfers were inspired by him.”
“Seve was in many ways the king of the recovery shot,” Wentworth’s Julian Small added.
“I always remember a wonderful shot — I think it may have been in 1991 — it was a very tricky shot and a very important shot in the game.
A little boy behind him was rattling his coins in his pocket and Seve stepped back and said to the little boy, ‘Are you nervous?' He said, ‘Yes'; Seve said: ‘So am I!'.”
The Seve Ballesteros Foundation, which Ballesteros set up following his diagnosis to help others with brain cancer, was the chosen charity for the BMW PGA Championship at Wentworth.
The club, where Ballesteros won five World Match Play Championships and the 1991 PGA Championship, will host a tribute event during this year's renewal of the latter.
Chief executive Julian Small added: “Along with other greats of the world of golf, Seve left his footprint here.
“We wanted to make a difference and during the week of the BMW PGA Championship, on the Monday we're going to hold a celebration of Seve's life and a celebration of golf. We'd like to think we can raise £300,000 for the Seve Ballesteros Foundation.”
Ballesteros had joined forces with Cancer Research UK to raise money for the charity to help fund research into brain cancer.
Today the chief executive of Cancer Research UK Harpal Kumar said: “We're very saddened to hear that Seve Ballesteros has lost his fight against brain cancer. I had the great honour of meeting Seve and his family at his home in Pedrena when we first established the Seve Ballesteros Foundation partnership in 2009 and my thoughts are with his family at this very difficult time.
“Seve was a hero to many for all he achieved during his career, but never more so than in the months after he was diagnosed with a brain tumour. His personal battle against the disease, and his determination to help others through his Foundation, was truly inspirational.
“Seve's experience brought a much needed spotlight onto brain cancer and Cancer Research UK was honoured to join forces with his Foundation to work towards our shared goal of beating brain cancer. It's a huge challenge but one that Seve was committed to making a reality.
“The partnership with Cancer Research UK has already raised over £700,000 towards ambitious and vital research that will help improve diagnosis and treatment of brain cancers in the future.
“Today we have lost a man who will be much missed and remembered the world over for what he achieved both on and off the golf course.”
Among hundreds of tributes by users of Twitter, Piers Morgan wrote: “RIP Seve Ballesteros — greatest genius golf has ever seen.”
Former England cricket captain Michael Vaughan wrote: “Not many can say they made the sport they played trendy, cool, fun and sexy. Seve single handedly did that.”
BBC sports presenter Jake Humphrey wrote: “Legend is the most overused word in sport. It applied to Seve Ballesteros. He changed the face of golf. Thoughts with his family. RIP”
The family's statement at www.seveballesteros.com read: “At 02:10 hours [Spanish time], Seve Ballesteros — accompanied by his family at his home in Pedrena — died due to respiratory failure.
“The family appreciates all the expressions of support and affection they have received since Seve was admitted on October 5, 2008 at the Hospital Universitario La Paz in Madrid.
“At the same time, please respect our privacy at such a painful time. Thank you very much.”
Ballesteros, who announced his retirement from golf in 2007, recevied confirmation that he had a brain tumour a year later.
But in December 2009 he appeared on television to receive the BBC's Lifetime Achievement Award at the Sports Personality of the Year event from his former Ryder Cup partner - and now captain - Jose Maria Olazabal.
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