Cheshire police in north-west England, where Speed lived, said in a statement that a 42-year-old man had been found hanged at his home and that there were no suspicious circumstances.
Tributes poured in from hundreds of colleagues and former team-mates while thousands of fans made their feelings known when they burst into spontaneous applause during a minute's silence in Speed's memory before the start of Swansea City's match with Aston Villa in the Premier League.
The FAW said in a statement: "That this tragedy should have overtaken someone so young and talented is a huge loss not only for his family and friends but a nation as a whole.
"We extend our sympathies and condolences to the family. We ask that everyone respects the family's privacy at this very sad time."
Speed, who was made a Member of the British Empire (MBE) for services to football last year, leaves a wife and two sons.
Wales First Minister Carwyen Jones said the news was "devastating" and that he was deeply saddened, adding: "our thoughts are with his family at what must be a very difficult time for them."
Speed played for Leeds United, Everton, Newcastle United, Bolton Wanderers and Sheffield United in a playing career that spanned 22 seasons and he became Wales manager last December.
He was a member of the Leeds team that won the last First Division title in 1992, before the start of the Premier League, and he became the first player to make 500 Premier League appearances during his time at Bolton in December 2006. He retired after making 535 appearances in the Premier League.
He also made 85 appearances for Wales between 1990 and 2004, becoming their most capped outfield player and second highest overall behind goalkeeper Neville Southall.
He then turned to management - briefly with Sheffield United as player-manager then with Wales from last December. Although Wales were eliminated from the race for the Euro 2012 finals, they had won their last three internationals and there were signs of real improvement under Speed's guidance.
MINUTE'S SILENCE
The news of his death broke a short while before Welsh club Swansea were due to play Villa at Swansea's Liberty Stadium and after some debate by the authorities about whether the match would go ahead or not, the decision was taken that it would.
Four of Speed's Welsh squad started the match: Ashley Williams, Neil Taylor and Joe Allen of Swansea, with their international team-mate James Collins lining up for Villa.
Speed's former Newcastle United team-mate Shay Given, playing in goal for Villa, sobbed openly before the match began.
A minute's silence before kick-off was broken with the crowd bursting into applause after a few seconds and singing "One Gary Speed, there's only one Gary Speed." The match, played in a subdued atmosphere, ended in a 0-0 draw.
"He was enjoying outstanding success as Wales team manager and our thoughts are also with those Welsh internationals on duty today for both Swansea City and Aston Villa," the stadium announcer told the crowd.
Speed's former Wales team-mate Robbie Savage was among hundreds paying their respects on Twitter, saying: "The world has lost a great man in Gary Speed. I am devastated. Spoke to him yesterday morning. Why why? I will miss him so much.
"He was upbeat on the phone, yesterday we were laughing together, talking football and dancing. He was a great team-mate and a great friend. RIP."
Howard Wilkinson, who knew Speed as a teenager and was manager when he was in the Leeds team that won the title in 1992, told Sky: "It is tragic, it is unbelievable. It is such a loss.
"He was such a great, great bloke, I cannot believ
