And I can't think of a whole lot of anime that's actually scary, or anything else you would want from serious horror. There's plenty of horror-themed shows out there, but most of them fall prey to anime tropes that just make them your average high school drama, or just aren't really scary or creepy or whatever, which is weird when you consider just how bizarre and ****ed up Japanese horror can be. But anime is generally made for a younger audience, so it just doesn't get the same J-horror treatment as more adult movies do.
I hear things about a show called Another (but I haven't seen it so I can't say), and there's a manga (Japanese black and white comic) called Uzumaki which, from what I've read, is pretty surreal and creepy, but I've yet to find an anime that really feels truly great as far as being legitimate horror.
Perfect Blue is actually a pretty great psychological thriller, and it's surreal enough that it can scratch your horror itch.
Waitaminute, I have seen Another. It's pretty much a more fleshed out interpretation of Final Destination, but it's actually a pretty solid horror show. Not so much about scares as it is about the way that people interact in stressful situations, it's still a high school drama, but one that is taken seriously. I imagine the school setting is just one of those things that an anime studio demands, cause it's just the way that the anime industry works, but it's a legitimately good and unsettling show. I wouldn't call it brilliant, but its juxtaposition of high school kids being high school kids, and an atmosphere of impending death around every corner, is pretty compelling, especially since it doesn't shrink away from some rather disturbing imagery, which is pretty rare for TV anime.
NSFW
Oh, and you might recognize this panel from Uzumaki as one of Exo's avatars. That's right, Exo is secretly an anime dork, cause only true anime dweebs read manga.