Abstract: This talk aims to provide an accessible introduction to the finite element exterior calculus, or FEEC, a new approach to designing and understanding numerical methods for a variety of PDEs. FEEC arises at the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computing, the other from topology and geometry. The former stream provides key concepts like Galerkin methods, saddle point variational principles, and finite elements, while the latter contributes ideas such as elliptic complexes, de Rham cohomology, and Hodge theory. These tie together elegantly in the theory of Hilbert complexes and their discretization, which can be used to guide the development of new finite element methods. FEEC has succeeded in unifying, improving, and extending finite element methods for a variety of PDEs arising in fluid and solid mechanics, electromagnetics, and other areas, and has led to the discovery of simple stable numerical methods for problems which were previously intractable.