A descendant of both Judge Roy Bean and Daniel Boone, Avery came to California from Texas in the early 1930's in search of a cartooning career. Learning the animation process as an inker, painter, and storyboard artist for Walter Lantz, Avery joined the Leon Schlesinger Studio as a director in 1935. Leaning away from imitating Disney cartoons every other studio in Hollywood was producing, Avery and his colleagues perfected the gag cartoon, with sophisticated humor that took audiences by surprise. He made fun of animated cartoon traditions, lampooning fairy tales and creating off-beat, wise-guy characters.

Avery introduced Daffy Duck in 1937, and directed "A Wild Hare" in 1940, the cartoon which crystallized the personality of Bugs Bunny.

Avery moved on to MGM in 1941, and later produced television commercials. His legacy was the inspiration for the zany course that Warner Bros animation would take.