I only saw Danganronpa: The Animation so far. And I haven't played any of the games yet.

Oh no, don't watch it in that case.
Danganronpa: The Animation was terrible compared to the game.
The new anime focuses on events (and people) you won't be familiar with without playing the second game.
I would recommend playing both of course. ;)

Oh, alright. Also I found this useful thread on reddit.

Re:Zero - It starts out as another one of the guy-gets-teleported-to-a-fantasy-world shows, but gets really dark fairly quick. Toss in a Groundhog Day-esque mechanic just for added fun. It's good and I want to see where the story is going, but damn.... It's up to episode 16 on Crunchyroll.

Dunno if listed in the thread but you should watch:

Neon Genesis Evangelion- absolute mindfuck
Cowboy Bebop- good scifi
Gundam Wing- also good scifi
Sailor Moon- if the power rangers were schoolgirls
Rurouni Kenshin- history
Samurai Champloo- history + rap/music

91 Days is very awesome. The series is about a young man who lost his family to a power struggle between mafia families when he was still a child and is now infiltrating the responsible family plotting revenge. There are only two episodes out by now but I think I can already say this is one of the best series in the last few years. The time and place is a small backwater town in the USA during the prohibition which is the perfect setting for a mafia story.

D: I thought the series had stopped at 13! Downloading :D

I can confirm Steins Gate Movie is brilliant. :)

Soon not bad.

Did you step out of a time machine? Or are you just listing the classics?

Those are the only anime I could really recommend, along with Digimon, Pokemon, and One Piece, but I figure people have seen the former three.

The only ones? O_o

Ergo Proxy, which will soon (August) be available in a limited Blu-ray edition with Italian dub audio. Finally...

Moe has twisted anime for sure but theres plenty old and new worth watching.

Trigun, Berserk, Full Metal Panic, FLCL, Rahxephon, Ghost in the Shell, Hellsing, Serial Experiments Lain, Psycho-Pass, From the New World, Kino's Journey, Basilisk, Haibane Renmei, Last Exile, Requiem from the Darkness, Steins;Gate, Code Geass, Boogiepop Phantom, Excel Saga, Akira, Strawberry Marshmallow, Mushishi, Battle Angel, Jin-Roh, Bodacious Space Pirates, Fate/Stay Night, Heroic Age, The Twelve Kingdoms, Claymore, Samurai 7, Gunslinger Girl, Death Note, Paranoia Agent, Gantz, When They Cry...etc...
There are just a few titles that I actually own physical copies of. There is plenty more but these are more universally popular titles.

Didn't bother listing Studio Ghibli titles. They're required watching for anybody weather they are a anime fan or not.