A useful approach to the study of fusion categories is to understand them up to a Morita equivalence. It turns out that two fusion categories are Morita equivalent if and only if their Drinfeld centers are equivalent as braided categories. Thus one is led to analyze the structure of such categories. In this talk I will introduce the new invariant of a braided fusion category C (called the core) which separates the part of C that does not come from finite groups. I will describe properties of the core and present some classification results. This talk is based on joint works with S. Gelaki, V. Drinfeld, P. Etingof, and V. Ostrik.