Calabi-Yau manifolds play an important role in differential geometry and theoretical physics. Geometrically, such manifolds can be characterized as being self-dual in a certain natural sense. This self-duality property admits a vast generalization leading to the notion of a Calabi-Yau algebra and, more generally, Calabi-Yau category. In recent years, Calabi-Yau categories found applications in many areas of mathematics: representation theory, algebraic geometry, symplectic topology, string topology, conformal field theory (to name a few). In this talk, after giving a gentle introduction to Calabi-Yau categories, we will discuss some of these applications as well as some of our recent work.