Oooh anime... I'm not sure I like any specific genres of anime because the ones I like are kind of all over the place genre-wise. But some of my favourites are Ano Hana, Haibane Renmei, Tsubasa Reservoir Chronicle, xxxHolic, Clannad, Angel Beats!, Cardcaptor Sakura, Soul Eater, Dennou Coil, Durarara!!, Madoka, Cowboy Bebop, Ghost in the Shell, Code Geass, Ouran High School Host Club, Steins;Gate, Samurai Champloo, Claymore, Twelve Kingdoms, FLCL, Princess Tutu, Evangelion (original series and Reboot movies), Fullmetal Alchemist (both the first series and Brotherhood), Baccano!, Sailor Moon, the Persona animation. Movie-wise I loved The Girl Who Leapt Through Time, Grave of the Fireflies and Summer Wars. And I also basically love every Ghibli film but special mentions to Princess Mononoke and Howl's Moving Castle.
I've also watched Naruto, Bleach, Elfen Lied, Death Note, Blood+, D. Gray Man, Vampire Knight, Black Butler, Gundam (like, all of them LOL)... uhh, a bunch of others, but they're not among my favourites. I used to like Naruto and Bleach a lot when I was younger, but not so much anymore.
Oh man, looking at that list, I realise I must watch a lot of anime!? Wow.
ETA: Oh, and Attack on Titan, Free! and Kill La Kill are on my to watch list! I don't know if they'll be my thing, but everyone's talking about them, so.