Arians replaces the fired Dirk Koetter, who spent three seasons as the Bucs head coach and agreed Tuesday to become the Atlanta Falcons' offensive coordinator.
Licht was vice president of player personnel during Arians' tenure as head coach with the Cardinals.
"His willingness to work is unbelievable", Arians said.
During his five seasons with the Arizona, Arians recorded a combined 40-27-1 regular-season record along with a post-season one of 1-2.
Bowles played for Arians at Temple before two successful seasons as Arians' defensive coordinator in Arizona.
Arians is a two-time Associated Press NFL Coach of the Year.
During Koetter's stay with the Bucs - first as offensive coordinator, then as a first-time National Football League head coach who retained play-calling responsibilities for most of his tenure - Tampa Bay posted four of the five most productive offensive seasons in club history. According to the Tampa Bay Times, "Arians reportedly did not bring a doctor's note to his interview with the Bucs".
A new coaching staff's goal: seeing if Jameis Winston is the quarterback the Bucs can build around as he enters the fifth and final season of his rookie contract. Since 2017, however, he's missed eight games and nine starts, failing to convince Koetter in 2018 that he belonged as Tampa's full-time starter.
Among the high-profile quarterbacks he has worked with during a long coaching career are Peyton Manning, Ben Roethlisberger, Carson Palmer and Andrew Luck.