Explore the classification of unitary modular tensor categories (UMTCs) in this comprehensive lecture from the Graduate Summer School on Mathematics of Topological Phases of Matter. Delve into the algebraic models that capture the properties of topological phases of matter in 2D, and discover how a classification of UMTCs could provide a periodic table-like framework for these phases. Review current knowledge, open problems, and successful techniques from category theory, algebraic geometry, number theory, and representation theory. Gain insights into the pursuit of classifying UMTCs from various perspectives and understand their significance in the field of topological phases of matter.