Strachan, who took charge of 40 games since being appointed in 2013, failed in his bid to take Scotland to a first major tournament finals since the 1998 World Cup.
"The Board of the Scottish FA and Gordon Strachan have agreed that the Scotland national team will be led by a new head coach in preparation for the UEFA EURO 2020 qualifying campaign," a statement from the SFA said.
Scotland missed out on a playoff place for next year's finals in Russia by failing to beat Slovenia in their final Group F qualifier on Sunday.
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"I share the profound disappointment at missing out on the playoffs, especially having worked so hard to fight our way back into contention," Strachan said.
Strachan won 19 of his matches in charge.
Assistant coach Mark McGhee is also leaving his post.
(Reporting by Martyn Herman; Editing by Toby Davis)
