Wright, previously the Potters' academy manager and head of coaching and coach development, succeeds Terry Robinson, who left the club this summer to take up a role as Leicester's director of football.
Wright told www.stokecityfc.com: "I am very much looking forward to the challenge.
"It's a great achievement what we have done so far and the important thing now is to maintain that category one status, but also at the same time ensure that we are in a position to produce good young footballers."
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