Snodgrass, who had six months left on his West Ham deal, is manager Sam Allardyce's first signing at the Hawthorns as the club look to climb out of the Premier League relegation zone.
West Brom are second-bottom in the league standings, with eight points from 17 games, and Allardyce hopes the Scot's experience can help steer his side to safety.
"He'll add Premier League experience and Premier League quality to our side," Allardyce said of Snodgrass.
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"He's left-footed. We all look for balance. When you get the right balance in the team with left-footed players, they can really produce for you down that left side.
He tells me he's played holding midfield, left of a three, wide left and right, so he's pretty versatile across midfield.
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Snodgrass scored 11 goals and provided 17 assists in 86 games for West Ham.
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