Conway gained a following from his appearance in the 1960s sitcom McHale's Navy. Afterwards, he starred in a string of doomed series before appearing in several slapstick family films. Of these films for Disney, he was often paired with fellow funnyman, Don Knotts. The most popular of these is probably The Apple Dumpling Gang series of movies.
He is probably best known, however, for his work on the Carol Burnett Show where he sometimes caused his fellow players to fall out of character by busting out laughing from his antics. Conway's work on the show earned him three Emmy Awards.
Conway continues to appear in movies and has cameo appearances in TV series. Most of these appearances showcase his comedic talent. Currently, Conway voices the character "Barnacle Boy" in the popular Nickelodeon cartoon series Spongebob Squarepants.