It is usually about avoidance, not danger
Being chased most often dramatizes something you are running from while awake: a difficult conversation, a deadline, a feeling you would rather not sit with, or a decision you keep postponing. The terror in the dream is the emotional weight of the thing you are avoiding, made vivid. The faster and longer you run, the longer the issue has likely been deferred.
Who or what is chasing you matters
An unknown figure often represents a part of yourself you have disowned — anger, ambition, grief. An animal can point to an instinct or appetite you are trying to suppress. A specific person may reflect an unresolved dynamic with them, or a quality of theirs you see in yourself. Paying attention to the chaser is usually more useful than the running.
When you turn and face it
People who learn to stop and confront the pursuer in a recurring chase dream frequently report the dream changes or stops — a striking mirror of what happens in waking life when an avoided thing is finally addressed. The chase tends to dissolve the moment it is met directly.