It's true that anime series like "Cowboy Bebop" have elements never considered in 1950's TV, like a definitive end to the series, foreshadowing and tragedy. But the mood of "Bebop", its music, its eccentric characters and the cynical humor of the hero can all be traced to "Peter Gunn." (And to show that nothing is completely original, some have said that "Gunn" was derived from Will Eisner's classic comic strip character of the 40's and 50's, "The Spirit.")
Gunn had a great supporting cast. There was the old jazz lady Mother, whose jazz bar just happened to attract the best West Coast jazz artists of the day (occasionally mentioned by name in the episodes); her house singer Edie Hart, whose love for Gunn was remarkably passionate; and Lieutenant Jacoby, who had a love/hate relationship with Gunn. There were equally weird characters involved. One episode in the second DVD volume has Gunn protecting Timothy - who happens to be a sea lion, with his own cute little theme song. More typical, in the first volume, was a story about a dead body found in Edie Hart's apartment, which is being painted. The attitude of the painter of all these police and goons in the apartment, and making his job harder, goes beyond comic relief to a featured comic part.