Trossard has agreed a four-year contract at the Amex Stadium.
The 24-year-old captained Genk to the Belgian title last season, scoring 22 times and contributing 11 assists in 47 appearances in all competitions.
Albion head coach Graham Potter told the club website: "We are delighted to welcome Leandro to the club and excited about the impact he can have for us.
"He's an attack-minded player who had a very impressive season last year in helping Genk win the league title.
"He was influential in the side's success last term and scored a lot of goals, as well as creating chances for others.
"At 24 he also brings good experience, having captained the side and also featured many times in the Europa League."
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