The fear of being chased in dreams stems from real life anxieties. Have you committed an atrocious act that you are hiding from everyone and scared of being found? You are running away at the same time being chased in your dream, because it is instinctive to avoid being caught. There is a lot of pressure on you to keep avoiding the issues at work or your relationships.
The person being chased in the dream may be trying to avoid an accident, escape death, or get away from a situation where they feel trapped. Generally speaking, if someone is chasing you in a dark alleyway then it might imply that they are after your money, whereas if they are chasing you across a field then they may be interested in your affections or attention.
If you continue to have these being chased dreams very often, seek out a person you can trust and discuss with them how you can confront the issues at hand. There is no need to keep feeling vulnerable every night you fall asleep.