Vector-valued Modular Forms

We give a short overview of the approach to the theory of vector-valued modular functions and forms developed in the past few years in collaboration with T. Cannon. In particular, we present the notions of fundamental and characteristic matrices, discuss the tools available for their computation, and explain the trace and spectral conditions. The relation to Jacobi forms, and the extension of the theory to subgroups of the modular group are indicated, and a section is devoted to the invariant differential equations satisfied by specific forms. A general dimension formula for vector-valued modular forms is also presented.