Adams was born in Manhattan, son of William Yarmy and his wife Consuelo (Deiter). Adams and his brother Richard (who later became an actor, known as Dick Yarmy) were each raised in the religion of one parent: Don in the Catholic faith of their mother, and Dick in the Jewish faith of their father. Dropping out of New York City's DeWitt Clinton High School (comedian Larry Storch was a classmate), Adams worked as a theater usher.

Adams was the voice of the title character in Tennessee Tuxedo and His Tales (1963-66), with his bombastic catch-phrase "Tennessee Tuxedo will not fail!" and later was the voice of Inspector Gadget in the initial run of that television series (1983-86) and the Christmas special as well as in later reprises such as Inspector Gadget's Field Trip. He retired from voicing Inspector Gadget in 1999. later, he voiced himself in animated form for a guest shot in an episode of Hanna-Barbera's The New Scooby-Doo Movies, titled "The Exterminator". Starting in 1982, Adams resurrected the Maxwell Smart character for a series of television commercials for Savemart, a retail chain that sold audio and video equipment. He also appeared in the film Jimmy the Kid (1982) and played a cameo role as a harbormaster in Back to the Beach (1987). He attempted a situation-comedy comeback in Canada with Check it Out! in 1985; the show ran for three years in Canada, but it was not successful in the United States. The show also starred Gordon Clapp, an unknown actor at the time, who developed a rapport with Adams. Adams reprised his Maxwell Smart role on Get Smart for Fox in 1995, which co-starred Barbara Feldon and rising star Andy Dick as Max and 99's only son. Unlike the original version, this show did not appeal to younger viewers, and it was canceled after seven episodes. He later went on to voice the character of Principal Hickley in the late-1990s/early-2000s Disney cartoon, Pepper Ann. Adams was the voice of Brain the Dog in the end credits for the film version of Inspector Gadget in 1999, his last voice role before his death six years later.

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