Abstract: Going beyond the setting of convex cocompactness, there is an effort to develop a theory of geometric finiteness for subgroups of mapping class groups that captures a broader range of behaviors and relates these to the structure of Teichmuller space, the action on the curve complex and the geometry of surface group extensions. This talk will explain some of the motivation and goals of the theory and introduce recent examples and constructions. These examples include lattice Veech subgroups, which are perhapd the most compelling examples for geometric finiteness, as well as certain combination and right-angled Artin group constructions. This includes joint work with Matthew G. Durham, Christopher J. Leininger, and Alessandro Sisto and well as with Tarik Aougab, Harry Bray, Hannah Hoganson, Sara Maloni, and Brandis Whitfield.