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Richards moves to the Bundesliga champions from Reading, having made 42 appearances for the Championship club during the 2020-2021 season.
The left-back, who represented England at under-21 level, has agreed a four-year deal that runs through until 2025.
"My move to Bayern is a great honour for me. A dream has come true," Richards told Bayern's official website.
"I'm proud to be wearing the shirt of one of the world's biggest clubs. I hope I can help the team continue to be successful in the future.
"My thanks to the management at Bayern for their trust. Our conversations were very convincing. I can't wait to be on the pitch for Bayern."
Bayern sporting director and board member Hasan Salihamidzic is delighted with the latest addition to the squad as the club prepare to begin life under new head coach Julian Nagelsmann.
They had already completed a deal to bring in Dayot Upamecano from RB Leipzig, strengthening a defensive unit no longer able to call upon David Alaba and Jerome Boateng.
"We're delighted Omar is coming to Bayern," Salihamidzic said.
"Omar is a technically gifted player on the left side of defence. He finds good solutions going forward, he's very alert and we trust him to play a good role in our team."
Richards made over a century of appearances for Reading after coming through their academy. The English club confirmed they did offer the 23-year-old a new contract, but he has instead opted to continue his career in the Bundesliga.