The winger is the club's fourth signing of the summer, following the arrivals of Alex Mowatt, Quevin Castro, and Matt Clarke.
Reach has played 231 times for Wednesday, but he finally departed the club after their relegation from the Championship.
"It's clear that West Bromwich Albion is a very ambitious club." Reach said.
"I share those ambitions of wanting to play in the Premier League.
"It's the most exciting challenge of my career and it's certainly the biggest challenge of my career.
"I pride myself on making a lot of appearances for clubs. I want to make plenty of appearances here and have my best years here. I just's cant wait to get started."
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