In this talk, we mainly discuss limit sets, particularly omega-limit sets and attractors which are invariant on dynamical systems. We first deal with envelope theory related to IFS and its applications. We also discuss the invariant notions on 3-manifolds with a certain cohomology condition. For surfaces, a topological characterization of the omega-limit sets for analytic flows was solved up to homeomorphisms. In this regard, we give a natural generalization to the higher genus of the solution for the genus 0 case. We derive various applications to the descriptions of dynamic notions induced from the omega-limit sets on the spaces. These results are joint works with J. Choy and S.-H. Ku.