The show focused on the Huxtable family, an upper-middle class African-American family living in a brownstone in Brooklyn Heights, New York, at 10 Stigwood Avenue.[3] The patriarch was Heathcliff "Cliff" Huxtable, an obstetrician. The matriarch was his wife, attorney Clair Huxtable née Hanks. Despite its comedic tone, the show sometimes involved serious subjects, such as son Theo's experiences dealing with dyslexia, inspired by Cosby's child Ennis, who was also dyslexic.

Cosby had an unusually high level of creative control over the show. He wanted the program to be educational, reflecting his own background in education. He also insisted that the program be taped in New York City rather than Los Angeles, where most television programs were taped.

The earliest episodes of the series were videotaped at NBC's Studio One facility in Brooklyn. The network later sold that building, and production moved to the Kaufman Astoria Studios in Queens.