Wicketkeeper-batsman Ben Brown will captain Sussex in the Championship and One-Day Cup in 2018.
"Everyone that knows me knows how passionate and proud I am to be a Sussex player, so to become club captain is a special day for me and my family," said the 29-year-old.
Former England international Wright will lead Sussex in the T20 Blast.
Head coach Jason Gillespie added: "I know Ben would have loved to have taken charge in all three competitions, but we need to make sure we manage his workload as best we can.
"As a batsman and wicketkeeper and our club captain, Ben plays a number of massively important roles in the squad, and so it was felt that by asking Luke to captain the T20 side, this would make sure that Ben was not overloaded.
"We're very lucky to have someone of Luke's calibre to take the reins in the short-form. He has vast experience of the game all over the world, something that will be of tremendous value in leading the T20 side."