The first four Batman films -- "Batman," "Batman Returns," "Batman Forever," and "Batman & Robin" -- are completely unrelated to the Nolan films, and the Nolan films are NOT prequels to those other films. Those films exist in their own separate reality that is not part of the comics, either.
In addition, for fun I wanted to add an element to the comics that served as inspiration for the Nolan films. "The Man Who Falls" is a Batman comic story by Denny O'Neil from 1989, which basically tells Bruce's history and his quest to become Batman.
It begins with him falling down the hole into the cave where bats scare him, followed by the depiction of his parents' deaths, and then Bruce tries college but drops out, and finally loses faith in the justice system and determines to find a way to work outside the system. This leads him to Asia, where he trains in a mountain monastery to develop his physical and mental prowess, but is told he is still too dominated by his fear and trauma. He leaves and begins to train with Henry Ducard, who trains him further in the martial arts as well as stealth and the use of deception and cunning. But Bruce has a falling out with Ducard when Ducard executes a criminal. The themes about "falling down" and picking yourself back up from failure are repeated in the story, and it also goes on to show Bruce in the snowy mountains (during detective training) where another of Bruce's trainers falls off the edge of a snowy cliff.
If all of that sounds pretty familiar, it's because it is a fairly close parallel to many elements during the origin portion of "Batman Begins." But a lot of people don't realize this story even exists, let alone how significant it was as an influence to the films.
Another series of Batman comics that was a clear influence -- and that is a very controversial subject -- is Steve Englehart's "Dark Detective" stories (divided into three separate parts). It included a new blond, handsome politician running for office, who has become the new love interest and fiance of Batman's former girlfriend, Silver St. Cloud. Bruce and St. Cloud are still in love, but she could not be with him because he is Batman -- but now they grow closer again, and she is going to leave the politician to be with Bruce again. The blond politician is a hero of Gotham, bravely standing up against corruption and crime. But the Joker kidnaps St. Cloud, and the politician tries to save her, causing him to have his arm and leg cut off on one side. Depressed and having lost Silver St. Cloud, the once proud and courageous blond politician gets a visit from Two-Face, who has a talk with the politician and convinces the man that life is cruel, and that he should embrace his dark side and lash out at the world. This pushes the politician over the edge and into evil.
Granted, this arc is spread out over a lot of books and separated into three distinct runs that are divided by several years, and the last run was written out but never went to publication. Still, the arc is clear and a major element of the stories, and seems to be a clear inspiration for some themes and elements in "The Dark Knight."
So while the Nolan films are definitely all on their own apart from the comics and not in any way canonical, they do at least have a lot of their inspiration from the comics, so if you look hard enough you can see where the comic book canon has had its impact on the films, and why these films are really faithful to the best elements of the source material. I guess you could say, then, that while the films don't "fit into the overall Batman canon," the Batman canon has in many ways fit into the films.
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