Bosley was born in Chicago, Illinois and during World War II he served in the U.S. Navy. While attending De Paul University in Chicago in 1947, he made his stage debut in Our Town with the Canterbury Players at the Fine Arts Theatre.
His breakthrough stage part was as New York's Mayor Fiorello H. LaGuardia in the long-running Broadway musical Fiorello (1959) for which he won a Tony Award.
He is best known as Ritchie Cunningham's father, Howard in the long running sitcom Happy Days.