The theory of majorization was introduced by Hardy, Littlewood and Pólya in order to formalize the intuitive idea of one set of numbers being more "spread out" than another. They established a surprising characterization of this property in terms convex functions, which allowed them to provide a unified approach to a number of seemingly disparate inequalities from in the literature from that era. The theory of majorization has subsequently found important applications throughout mathematics, mathematical economics and, more recently, quantum information theory. In this talk, I will discuss these developments and introduce a generalized theory of majorization, where numbers are replaced by (not necessarily commuting) matrices. This is joint work with Paul Skoufranis.