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Even though anime has made its way into the mainstream over the past few years, negative notions about the medium persist.
With the exception of a few mature animated comedies (some of which are fantastic and thematically complex, like Bojack Horseman), animation in the US is still typically viewed as a medium for children. The idea that cartoons are kid sh*t, while perhaps understandable for someone who has only ever been exposed to Western media, is ignorant of the broad range of animation in other cultures.
In a similar vein, a lot of people insist that they just can't get into anime, or they stigmatize all of it because they don't like the "schoolgirl stuff." But it's important to remember that anime isn't a genre–It's a medium.
Individual anime series fall into every genre under the sun, just like movies and live-action TV shows. Saying you don't like anime because of the schoolgirl stuff (which is a very valid thing to dislike) is kind of like saying you don't like movies because of slasher films. You're writing off an entire medium of art over a genre that you can easily avoid.
So let's say you are open to watching anime, but aren't quite sure where to start. Or, more likely, maybe you already love anime and you're trying to find a series to convince your SO that the $200 action figure in your room was a totally reasonable thing to buy (it was, and your life choices are perfect). Just check out any of these gateway anime series that serve as perfect entry points into the medium's diverse offerings.
One Punch Man
One Punch Man is probably best appreciated if you already have some experience watching other anime (it subtly plays off of a lot of other series and tropes), but it's also simple and light enough to enjoy without prior knowledge. The series is a satire on the classic shonen trope of a protagonist who wants to become the best (INSERT: Pirate, Ninja, Warrior).
In One Punch Man the hero, Saitama, has already become the strongest fighter in the world, so strong that he can defeat any enemy in one punch. The problem now is that he's still entirely unfulfilled. It's very funny, and offers some surprisingly deep insights into depression, too.