Conti was on the Italian national side that won the 1982 FIFA World Cup. He was one of the best players in the tournament, and played an instrumental role in Italy's win in the final against West Germany. Despite being booked after 31 minutes, he was a central figure in Italy's second goal that was scored by Marco Tardelli and single-handedly created Italy's third goal by breaking down the right side from the half-way line and crossing to Alessandro Altobelli to score from the top of the box after 81 minutes. The Italians won the game 3-1.
During his domestic league career, he played for Roma from 1973 to 1990, winning the Scudetto in 1983 and Coppa Italia four times. He helped them to the 1984 European Cup Final but were defeated on penalties by Liverpool, with Conti missing his penalty in the shootout.
He is one of eleven members of Hall of Fame of A.S. Roma.
Following the departure of coach Luigi Delneri, Conti moved up from his position as head of the club's youth teams to first team coach during the 2004-05 season. Conti did not possess a coaching licence, but was a respected figure at the club. World Cup winners, however, are exempted from taking coaching exams.
He led Roma to the Coppa Italia final and to a UEFA Cup berth. His short term role as caretaker manager finished when Roma appointed Luciano Spalletti as coach. His current position is club director of youth categories.