In this talk I will describe and illustrate a new approach which has added greatly to our understanding of stability of finite element methods and enabled the development of stable methods for previously intractable problems. This approach is homological, involving differential complexes related to the problem to be solved, discretizations of these complexes obtained by restricting the differential operators to piecewise polynomial subspaces, and commuting projections relating the two. The best known case is the de Rham complex, which underlies both electromagnetic and diffusion problems. In this case there are a large number of possible piecewise polynomial subcomplexes of each order. These can be presented systematically using the Koszul complex. The elasticity equations are related to another differential complex which can be related to the de Rham complex through a subtle homological construction, the discretization of which has led to new stable mixed methods for elasticity.