We will give a short overview of the Nielsen-Thurston Classification problem (classifying the homeomorphism type of surfaces) on finite-type surfaces and then move to infinite-type surface mentioning what is known and pointing out some difficulties. We will then discuss how to approximate, in the compact-open topology, a general self-homeomorphism of an infinite-type surface (joint with Y.Chandran) and potential definitions of pseudo-anosov mapping classes in the infinite-type setting (joint with F.Valdez).