Being chased is one of the most commonly reported dreams across every culture that's ever been surveyed on the subject, Korea included — which makes it a good example of a dream sitting at the intersection of a specific cultural tradition and a near-universal piece of human sleep psychology.
In Korean interpretation, the identity of the pursuer changes the reading substantially. Being chased by an animal is generally read through that animal's own symbolism layered on top — a chase by something already read as fortunate, like a pig, is interpreted more gently than a chase by something already read as threatening. Being chased by an unknown person or an unclear, formless presence is read more as a general warning about an unresolved problem or pressure in waking life, without pointing at anything specific.
What happens at the end of the chase carries real weight in the traditional reading. Successfully escaping, outrunning, or turning to confront the pursuer is read as a positive flip — overcoming a real obstacle or pressure you're currently facing. Being caught is read as a warning that the underlying issue is more serious or more urgent than you've been treating it as.
The cross-cultural, non-folklore explanation is worth sitting next to the traditional one rather than instead of it: chase dreams are strongly and consistently linked to ordinary waking stress and unresolved anxiety, which is very likely why the dream is so common everywhere, independent of any specific cultural tradition. A chase dream the week before a big deadline doesn't need ancient symbolism to explain itself.
Both readings can be true without contradicting each other. If you woke up from a chase dream, the honest, useful takeaway isn't "something bad is coming" — it's closer to "something is already pressing on you, and your sleeping brain noticed before you consciously admitted it." That's worth paying attention to either way.