Actually, there were two series that ran one after the other: Rocky and His Friends (1959-1961) and The Bullwinkle Show (1961-1964). Both were produced by Jay Ward, and featured essentially the same cast of characters.
Each episode contained several recurring elements: Fractured Fairy Tales, Peabody’s Improbable History, Dudley Do-Right of the Mounties, Mr. Know-It-All, Aesop and Son, and Bullwinkle’s Corner.
The good guys were, of course, Rocket “Rocky” J. Squirrel and Bullwinkle J. Moose. They foiled the plans of the bad guys, who were (of course) Natasha Fatale and Boris Badenov. The show all but developed the concept of dual interest – simple cartoon hijinks for the kids, puns and parodies of current events for adults. All the modern animated productions should be sending Jay Ward’s heirs royalty checks.
Close observers of the show have noticed lots of “continuity” errors on the show – Boris Badenov’s mustache would come and go, colors would change, clothing wasn’t the same. The reason is that the production was animated in Mexico to save money. It became clear that with cost savings came a sacrifice in quality. Some things don’t change, do they?