The newly-promoted Premier League side have forked out £10m for the services of the former England u-21 international, who has signed a four-year contract at the City Ground.
Richards joined Bayern Munich from championship side Reading last summer and contributed 12 appearances as the Bavarians won the Bundesliga and 17 appearances across all competitions.
An official statement from Julian Nagelsmann's side read: "FC Bayern have agreed to the transfer of 24-year-old Omar Richards to English Premier League club Nottingham Forest."
"The full-back joined the German record champions last summer on a free transfer from Reading in England’s second tier."
Bayern Munich sporting director Hasan Salihamidzic added: "It was Omar's wish that we allow him to join Nottingham Forest. He sees the opportunity to become a first-team regular at a Premier League club in his home country."
"Forest are getting a very good player and a lovely guy, who will forge ahead. We wish him all the best."
Richards is Nottingham Forest's fifth summer signing and their third acquisition from the Bundesliga.
Forest are back in the English top flight for the first time since 1999.
"I'm thrilled to return to England by joining Nottingham Forest at such an exciting time," Richards told Forest's official website.
"The club has enjoyed an incredible last few months and there's great ambitions for the Premier League return.
"I've played against Forest in the past and they get such great backing from their fans, whilst the City Ground is an amazing place that I can now call home."
