The 24-year-old defender, who previously had spells with Hull City, Aldershot and Portsmouth, is the 11th new signing made by Argyle this summer.
He has moved to Home Park after failing to agree a new deal with the Red Devils, after two years in Sussex.
Bradley played 49 games for Crawley last season, with his only goal coming at Argyle's fierce rivals Exeter City in a 2-2 draw on 14 November.
Plymouth have not disclosed the length of Bradley's contract with them.
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Plymouth Argyle have signed former Crawley Town captain Sonny Bradley.