The 59-year-old, who previously took charge for a short spell a decade ago, was officially unveiled on Friday morning following successful talks earlier this week.
Having now named McLeish as the successor to Gordon Strachan, the new man at the helm cannot wait to get going after nearly two years out of the game.
"It is a tremendous honour to be named Scotland national team head coach," he told the SFA's official website. "I am incredibly fortunate to have had the chance to lead my country already and I am immensely proud to be in this position for a second time.
"There is much to be positive about in Scottish football, with players playing at the highest level and promising talent breaking through.
"With Hampden Park as one of the host venues for Euro 2020 we have an amazing opportunity to showcase our talents to the world from our own back yard. I am relishing the challenge of leading us to a major tournament and I cannot wait to get started."
Scotland face Costa Rica in a home friendly on March 23, before heading to Hungary four days later.