The so-called "modern theory" of orbital magnetization was developed some 15-20 years ago. It takes the form of a reciprocal-space integral involving Berry curvatures and related quantities. A derivation based on semiclassical electron dynamics and another based on the Wannier representation led to identical expressions that could be implemented in modern first-principles electronic structure codes. In the first part of this talk, I will give a brief pedagogical review of these developments.
I will then discuss recent work in which we consider a crystal whose bulk magnetization vanishes by symmetry, but whose surface magnetization does not. An interesting question, then, is whether the surface orbital magnetization is well defined, and if so, how it can be calculated. Recently, Bianco and Resta derived a formulation of the orbital magnetization in terms of a local marker which, when integrated over the unit cell, correctly returns the bulk orbital magnetization. I will discuss whether, and under what conditions, a surface orbital magnetization is well defined, and point out ambiguities in the formulation in terms of a local marker. Calculations of surface magnetization and the resulting hinge currents will be illustrated in the context of tight-binding model Hamiltonians. I will also point out connections to the theory of the axion magnetoelectric coupling and surface anomalous Hall conductivity.