Finite element exterior calculus (FEEC) has had a transformative impact on the theoretical foundations of finite element methods. The fundamental paradigm is that differential complexes such as the de Rham complex are central to the analysis of partial differential equations in vector fields, and hence must be also be considered in the finite element discretization. This has led to a powerful theory of finite element de Rham complexes that now is the theoretical bedrock of numerical electromagnetism. This talk gives a broad overview of the theory and highlights several recent contribution: the extension of FEEC to manifolds, fundamental progress in approximation theory, and the numerical treatment of mixed boundary conditions.