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X-Men was in production at the same time as Batman, both shows premiered within a month of each other, so it's not so much that one was truly following the other. If anything, its both shows who helped spawn their clones. Like the DCAU shows, Spider-Man: The Animated Series follows X-Men and included a lot of the people who had worked on the earlier show, like Avi Arad.
Disney's Gargoyles has been said could have be influenced by Batman:TAS, with its similar Timm Style art design, dark and moody atmosphere, and complex storylines.
The success of Scooby-Doo launched a whole boatload of other series about mystery-solving/crime-fighting teens and their talking animal/car/whatever friend. Most of these copycats were actually produced by Hanna-Barbera, the same studio that created Scooby-Doo.
There are so many clones that Boomerang has a block called "Those meddling kids" dedicated to Scooby Doo, and its many, many clones.
And even before that, the clones and the original had their own team on Laff-A-Lympics, the Scooby-Doobies.
Even the Licensed Games inspired imitators in the form of Battletoads and Cheetahmen. Both tried to spin off cartoon series as well, but the Battletoads cartoon flopped instantly and the Cheetahmen cartoon never got off the drawing boards.
Whilst the extent to which more adult-orientated animated shows such as Family Guy and South Park are direct 'rip-offs' of The Simpsons is a subject of bitter and acrimonious debate across the Internet, it is fairly safe to say that without the enduring popularity of The Simpsons, which showed there was a market for animated programming aimed at more adult audiences, the former two shows — plus a lot of more obscure and more quickly forgotten similar shows — would probably have never been greenlit thanks to the Animation Age Ghetto.
After the show first premiered, there were a number of prime time animated shows that came out afterward, like Capitol Critters and Fish Police, that lasted barely one season.
Even Homer Simpson himself has been copied several times in shows like Bob in God, the Devil and Bob, Jay Sherman in The Critic, Peter Griffin in Family Guy which all feature a dumb and/or obese, pathetic slob as a titular character.
Teen Titans set the tone for the past few years of kids' action cartoons. Comedy-action blending and Rule of Cool became far more prominent, as did Animesque artwork (which was already gaining in popularity anyway).
This in turn caused Ben 10, whose success caused Cartoon Network to bring in more action cartoons to the network.
This looks to have gone back the other way—Teen Titans Go! looks to be an attempt to cash in on the DC Comics' success in a sitcom format. Over the 2010s, most of Cartoon Network's action shows have been canceled left and right in favor of surrealist programming.
It could be argued that even Spongebob Squarepants took more than its share of inspiration from Ren and Stimpy, minus the gross-out humor (at first, at least). The two shows share loose artwork, manic pacing, surreal humor, hand-painted close-ups of the characters, and the same music cues.
The Pre Cancellation episodes of Sponge Bob Square Pants are very popular, even to this day. Many cartoons have picked up the same formula where they are very joke driven as opposed to cartoons like Rugrats and Hey Arnold! & don't have plots that can relate to real life such as Adventure Time (The Exec Producer of that show is Spongebob co creator Derek Drymon), Regular Show, My Little Pony Friendship is Magic and Gravity Falls (with the only exception of this being Phineasand Ferb, despite the fact that it can be very surreal)
Parodied in The Simpsons episode "Flaming Moe's". The episode revolves around bartender Moe stealing Homer's idea for a drink recipe and renaming it the Flaming Moe. He even goes as far to rename his bar after the drink. His bar becomes a popular club spot, attracting even the likes of Aerosmith. After Homer outs the recipe to the general public, dozens of bars, restaurants and carts sprout up on the same street with variations on the name- e.g., Flaming Meaux.
During the mid-1990s, Cartoon Network produced a coupleof cartoon series where the characters had very stylized designs, intentionally looked rather one-dimensional and were drawn with thick black outlines around their bodies (the style was intended to be a throwback to certain 1950s cartoons). After the success of The Powerpuff Girls in 1998, this style became enormously widely used in cartoons both by Cartoon Network and by other companies, and remain so to this day, partly thanks to the rise of Flash animation.
This ironically mirrors how Limited Animation caught on in the 1950's. It started out as a unique artistic statement, but later became an excuse to create cheap, lazy animation.
Most prominently, and most obviously, is the new The Littlest Pet Shop, which shares a similar visual style and somewhat sarcastic but good-natured writing style. It hasn't achieved the huge success of Friendship Is Magic, but it does have its fans and is developing its separate identity.
The Jetsons was rather blatantly created by Hanna-Barbera to cash in on the success of their other animated sitcom in an unusual setting, The Flintstones.