The 47-year-old spent the last two seasons on the Indiana Pacers coaching staff after serving the six previous years with the Los Angeles Lakers under Phil Jackson.
He will take over from Coach of the Year George Karl, who was let go at the end of a season that saw the Nuggets eliminated in the first round of the play-offs.
"I'm very appreciative of the opportunity to lead this team," Shaw told the Denver Post. "Everything that's worth something, a lot of times you have to wait for it. I feel like I've waited and paid my dues."
Shaw played for seven different teams during a 14-year career and was a member of the Lakers' team that won three championships in a row from 2000-2002.
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- Indiana Pacers