Dreams of being chased are second only to dreams of falling as far as being the most universally experienced dream scenario.  Often, we have the same recurring dream of being chased from childhood through almost our entire adult life.  What is doing the chasing  - be it a monster, a mugger, or an animal of some sort - is  quite a personal thing and varies widely from individual to individual, so do make note of it because YOUR pursuer will definitely turn out to have unique significance for YOU.  In general, chasing dreams signify some sort of anxiety, fear, or feeling of entrapment in daily life, such as a stressful job or unhappy marriage.  Chasing dreams are literally "fight or flight" situations where you choose "flight."  These dreams don't occur if you have faced and dealt with whatever it is you fear during waking hours. 

Questions to ask yourself following a chasing dream follow:

Are you feeling entrapped, anxious, or powerless at home or at work?  Is your boss or co-worker overbearing?  Is your spouse overbearing, accusatory, or belittling?

Do you feel your job, relationships, home life, or social life is lacking?   Are you in a rut?

Are you subconsciously running away from something?  Have the bills become overwhelming?  Are the kids or spouse bearing down too hard?   Are you in a  bad relationship?  Do you feel guilty about something?

What is chasing you?  Is it a huge monster, a little spider - something unknown?   For huge monsters and the unknown, it's likely you are feeling the urge to get away from something, but you may not be able to put a finger on exactly what it is.  If it's a bug, snake, spider, etc., it may represent anxiety about some level of your life compounded by your innate fear of this particular creature, or it may be simply that you have recently had a close encounter with this particular creature and are working through your fears in your dreams.  If you are being chased by someone you know, look at your relations with this person.  Do you owe them money or are they making some sort of demand on you that you don't feel you can fulfill?

Can you get away from your pursuer?  If yes, then you are anxious for sure, but subconsciously you feel you can handle whatever the problem is.  If no, you likely feel there is no easy escape for the anxiety-provoking situation, and you likely feel trapped and helpless.  Many people report being chased in slow motion during dreams.  This represents your dream strategy to buy time to come up with a way out.  If you are completely stuck in the dream, it signifies that you are frozen inside with no idea which way to turn.  If you are stuck in quicksand and getting deeper the more you fight, you are realizing that there's no point in fighting your demon anymore.

Where are you being pursued?  Are you in an open field?  An enclosed maze?  The grocery store?   At home?  Enclosed spaces likely signify a strong sense of being trapped, whereas open spaces signify a feeling of overall vulnerability. 

Chasing dreams very often signify vulnerability, and the reason these dreams often recur over a lifetime is that certain situations trigger memories of childhood vulnerabilities.  When trying to interpret such a dream, look inward first - it may turn out to be your own demons haunting you.  Whether internal or external,  try to pinpoint what exactly it is that is making you feel insecure or threatened, and take positive steps to relieve the pressure.  Do write down any conclusions you come to and revisit that page in your journal in a few weeks to see if the issue has been reduced or resolved.