Folklore interpretations say that if you are looking at a chase or participating in it, you will have a comfortable old age. Although this may be comforting, there is a more realistic understanding of this activity in a dream.
If you are being chased, maybe you are running away from or trying to escape those things that are frightening and unpleasant (possibly your own habits and negative behaviors).
If you are doing the chasing, it may be that you are expressing some aggressive feelings toward others or are pursuing a very difficult goal. On the deepest level, if a stranger is chasing you it may represent your chasing a part of yourself, the unconscious attempts to catch up with the conscious in order for you to become more aware of yourself and your own multidimensional nature.