Raised in New York, Conaway was on Broadway at age 10, appearing with Colleen Dewhurst and Lillian Gish in "All the Way Home" (1960). He was 20 when he made his feature film debut playing a druggie in the heroin-soaked "Jennifer on My Mind" (1971). Conaway's big feature film break came with "Grease" (1978). While he had played the leading role of Danny Zuko on Broadway, he was had to cede that role to John Travolta and played the second male lead, the sexually-active tough guy Kenicke. But perhaps because of his active TV schedule, Conaway did not make another movie until "Covergirl" (1984) and subsequent film work was sporadic until the 90s when he starred in numerous low budget and direct-to-video productions, such as "A Time to Die" (1991) and "Almost Pregnant" (1992). He was associate producer of "Sunset Strip" and made his directorial debut with "Bikini Summer", both released direct-to-video in 1992.
Mostly though, Conaway is known for his small screen work. After making his debut in the syndicated "Sea to Shining Sea" in 1974, he began making guest appearances on numerous series, among them "Movin' On," "Happy Days" and "Joe Forrester". After leaving "Taxi", he landed the role of the heroic Prince Erik Greystone on "Wizards and Warriors" (CBS, 1983) and was the lothario who could barely keep his pants zipped on the short-lived "Berrengers" (NBC, 1985). As primetime offers diminished, Conaway moved to daytime joining the cast of CBS' soap "The Bold and the Beautiful" in 1991. Many of his TV-movies roles played on Conaway's often smart-alecky on-camera persona. Among the more notable were his turn as one of six buddies with women problems spending the summer together in the 1979 ABC miniseries "Breaking Up Is Hard to Do", a love-struck medical student opposite Deborah Raffin in "For the Love of It" (ABC, 1980) and a country singer kidnapped by two female convicts in "Nashville Grab" (ABC, 1981). He appeared in support of Joan Collins in "The Making of a Male Model" (ABC, 1983) and was one of the team of the revived "Dirty Dozen: The Fatal Mission" (NBC, 1988).
In 1990, Conaway made headlines when he sideswiped a bicycle rider on Santa Monica Boulevard and was arrested for drunk driving. The following year, he was ordered to 60 days house arrest and three years probation with mandatory attendance at Alcoholics Anonymous five days per week for three years.