The 44-year-old takes over from Chris Coleman, who led Wales to the Euro 2016 semi-finals but quit the job in November to take charge of English Championship (second tier) side Sunderland.
Giggs won 64 caps between 1991 and 2007 but was unpopular with some Wales fans for missing many internationals, especially friendlies.
He ended his playing career in 2014 and was Louis van Gaal's assistant at Old Trafford for the following two seasons before leaving United in 2016 after Jose Mourinho was appointed as manager.
"We are extremely delighted to see Ryan appointed as the national team manager," said FAW chief executive Jonathan Ford in a statement.
"With his vast amount of experience having played, coached and managed the biggest club in the world, we are confident of a successful future for our national team."
Giggs said he was proud to have been handed the job and excited by the challenge ahead, with Wales having failed to qualify for this year's World Cup finals in Russia and now focussing on the 2020 European Championship.
"I can't wait to start working with the players as we prepare for those crucial games later in the year," he said.
Giggs' first game in charge will be in the China Cup tournament against the hosts in Nanning on March 22.
(Reporting by Alan Baldwin, editing by Ed Osmond and Christian Radnedge)
