Coleman’s assistant Osian Roberts, former Wales striker Craig Bellamy and Mark Bowen were also interviewed.
Giggs had long been a target for the Football Association of Wales (FAW) and was the favourite for the role once Coleman departed after Wales failed to qualify for the 2018 World Cup.
‘I can’t wait to start’ – Giggs
The FAW confirmed the appointment on social media and in a statement Giggs said: “I am so proud to have been given the honour of managing the national team.
“The challenges that we have ahead of us with the Nations League and qualification for Uefa Euro 2020 excite me a great deal.
“I can’t wait to start working with the players as we prepare for those crucial games later in the year.”
FAW chief executive Jonathan Ford said: “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.”
‘Ferguson to blame for Giggs’ lack of Wales friendlies’
Giggs won 64 caps between 1991 and 2007 and captained his country – but he was criticised for regularly pulling out of squads and especially for missing friendly matches.
It is understood Bellamy and Roberts, who is the FAW’s technical director, also had their supporters at Welsh football’s governing body.
Bellamy, 38, won 78 caps and is the country’s fourth most-capped player.
The former Newcastle, Liverpool and Blackburn forward is currently an academy coach at Cardiff City, where he ended his playing career in 2014.Roberts has worked throughout all age grades in the Welsh system and been widely credited for helping to produce the talented crop of youngsters who will be at the disposal of the new senior manager and was Coleman’s assistant boss during a spell in which Wales reached the semi-finals of Euro 2016.
Tony Pulis was also among the initial contenders to succeed Coleman, but he ruled himself out when he became Middlesbrough manager last month.
Bowen was available after leaving Stoke, where he was assistant to boss Mark Hughes until the latter was sacked by the Potters on 6 January.