The finite element exterior calculus, FEEC, has provided a viewpoint from which to understand and develop stable finite element methods for a variety of problems. It has enabled us to unify, clarify, and refine many of the classical mixed finite element methods, and has enabled the development of previously elusive stable mixed finite elements for elasticity. Just as an abstract Hilbert space framework helps clarify the theory of finite elements for model elliptic problems, abstract Hilbert complexes provides a natural framework for FEEC. In this talk we will survey the basic theory of Hilbert complexes and their discretization, discuss their applications to finite element methods. In particular, we will emphasize the role of two key properties, the subcomplex property and the bounded cochain projection property, in insuring stability of discretizations by transferring to the discrete level the structures that insure well-posedness of the PDE problem at the continuous level.