There are plenty of genius female characters in comic books. You just don't hear about them as often because comics are pretty tough on dames. In fact, the medium's portrayals of women are just as antiquated and sexist as the term "dames." While male characters save the day, the women in comics are often cast as brain-dead bystanders or one-dimensional romantic prizes.
The women that do get to be heroes are trussed up as fetishized sex objects sporting bustiers and nylons when they should be wearing armor. Power Girl's costume even includes a boob window as if her designers forgot the industry-mandated gratuitous cleavage, only to remember right before an issue went to press.
Being shot through the spine by The Joker ended Barbara Gordon's career as Batgirl, but it didn't prevent her from fighting crime. As Oracle, she provides information, digital reconnaissance, and technological wizardry to support Gotham's many vigilante protectors.
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Dr. Harleen Frances Quinzel, M.D. was a psychiatrist at Arkham Asylum before encountering The Joker. She doesn't put that education to use much these days, but it's still locked away somewhere in that codependent and homicidal mind of hers.
#13 on The Best Comic Book Villainssee more on Harley Quinn
When superheroes need to go to court, they turn to She-Hulk for counsel. Not only is she capable of punching holes through mountains, she's one the best defense attorneys in the Marvel Universe, punching holes through any prosecutor's argument.
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Pamela Isley never got the chance to finish her degree in botanical biochemistry. Having your professor inject you with experimental toxins that turn you into a superpowered plant whisperer will do that to you. Nevertheless, she seems to have a pretty firm handle on the whole botany thing.
#27 on The Best Comic Book Villainssee more on Poison Ivy