The former Scotland international moves to the Liberty Stadium from League One side MK Dons.
He replaces Steve Cooper in the dugout and has signed a three-year contract.
Swans chief executive Julian Winter told Swansea' official website: "We are delighted with the appointment of Russell as our new head coach.
"He very much fits the profile of a young and hungry coach who wants to succeed.
"After earning plaudits with Milton Keynes Dons for the attractive, attacking style that he implemented, we quickly realised Russell would be the right person to take us forward.
"He captained Peterborough and Norwich, as well as winning promotion on five separate occasions (as a player), which tells you everything about his leadership qualities and winning mentality."
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