Why do I keep having dreams about being chased by zombies?

Chances are you've watched shows or read stories with zombies.  They are symbolizing something in your dreams, because you have introduced them into your mental lexicon.  From your experience, zombies represent something scary.  So, your mind conjures up zombies. An easy thing to try, for figuring out such dream issues, is to "ask" your zombie assailants why they are chasing you.  This can be done in your room, while you are awake, all by yourself.  *You* provide the voice for the zombie.  You ask it questions, and out loud, you start talking from the zombie's perspective. The reason this can work, is that everything in your dream is in your own mind.  You bypass your "this is too weird to make sense" editor, by doing it alone, and you "let the zombie talk."  For some reason, it usually works best if you actually talk out loud.  But you can try it inside your head, too. The process was originally created by Fritz Perls, and done where a psychologist asks you questions, and you act the parts of the dream.  But you can do it for yourself.  (Google Fritz Perls dream, for more info.)

Zombies are always looking to feed on life and usually accompany moments when one is lost, when one can't connect to an inner drive. Mind you, one may be quite convinced or determined about something one is doing, but one will or can invariably discover a deep misalignment with the deeper purpose of one's life. Externally it may manifest in various types of neurosis, excessive concern for what others do or think for instance...

Its probably a premonition.