First was Phantom - the Animation, probably a manga or video game adaptation. It's about a boy who travels to Los Angeles, where he ends up in a bad part of town. He is witness to a mob killing: a sniper shoots some mobsters just as he is passing. He runs away before the sniper can shoot him too, and ends up fighting her in an alley. He is saved by her supervisor, who sees a killer's moves in him. He gets drugged, and when he wakes up, he can't remember his old life. He gets trained to be a killer by the girl who nearly killed him, as an assassin in service to the mob.
It's a bit 'La Femme Nikita' and 'Gunslinger Girl', but with a male protagonist. You get the impression that the story only just begins after the first episode, but it's interesting enough to keep watching.
The other one was Interlude. The main character is a highschool student who experiences strange, life-like dreams and visions. Most of the time, he sees himself nursing his childhood friend (the hyper-genki Tamaki) who is terribly wounded. He dismisses these visions as dreams, but they become more and more vivid as time progresses.
Add a counsellor who sticks her hand through students and takes out some sort of wriggly worm, a bow-wielding disembodied shadow, a trio of civil servants and a girl with a demon-like dog, and you have something on the scale of 'Boogiepop Phantom'.
It's interesting to see where this goes (even though Tama-chan is rather irritating), and the designs are quite beautiful!