The 55-year-old was the overwhelming bookmakers' favourite to take the job after Craig Levein was sacked in November following a poor start to Scotland's World Cup qualifying campaign.
Strachan, who won 50 caps for Scotland, enjoyed a largely successful period as Celtic manager between 2005 and 2009 but had been out of management since leaving Middlesbrough in October 2010.
Scotland's chances of qualifying for the 2014 World Cup are effectively over but Strachan is aiming to finish the campaign strongly.
He said: “I am extremely proud to be appointed as Scotland national coach. There is a considerable challenge ahead in the World Cup qualifying campaign but we can take those first steps towards addressing that in the double header against Wales at Hampden and the visit to Serbia in March.
“Next month’s match against Estonia will give me an opportunity to assess the squad and, of course, it will be an unforgettable experience walking out as Scotland national coach at Pittodrie, a stadium that holds many special memories [for me] as a player.
“I can assure the supporters that I will do everything in my power to make sure we have a national team we can all be proud of and one that, in time, can experience the excitement of participating in a major championship finals once again.”
Strachan's first match in charge will be a friendly against Estonia at Pittodrie on February 6, before a World Cup qualifying double-header in March.
Scotland host Wales at Hampden on March 22 and face Serbia away four days later.
Strachan was a popular player for Scotland and represented his country at the 1982 and 1986 World Cups.
Stewart Regan, SFA chief executive said: “We are delighted to have appointed Gordon as the new Scotland national coach. As a player, his style and swagger made him a huge favourite of the Scotland supporters and few can forget his celebration against West Germany in Mexico in 1986.
“As a manager, he has experience at the highest level in England and Scotland, most notably taking Celtic into the last 16 of the Champions League. Within minutes of meeting Gordon to discuss the possibility, we were left in no doubt that he was the man to help rejuvenate our FIFA World Cup Brazil 2014 Qualifying campaign, starting with the March double header against Wales and Serbia.
“It is fitting that his first match as Scotland national coach will be a return to Pittodrie, a venue that holds many dear memories, and I am sure the Scotland supporters will be out in force to give him a warm welcome.”