The most natural formulation for the equations of elasticity is as a first order system, reflecting the very different nature of the equilibrium equation and the constitutive equation. Moreover this system applies more widely than second order formulations, for example to incompressible, plastic, or viscoelastic materials. The first-order system is captured variationally in the Hellinger-Reissner variational principle, which characterizes the symmetric stress tensor field and the displacement vector field as a saddle-point of a suitable functional. However it has proven extremely difficult to develop stable and effective finite element discretizations of this formulation--so called mixed finite elements for elasticity. Efforts to develop such methods go back to the earliest days of the finite element methods. However, stable mixed elasticity elements using polynomial shape functions have only been developed recently using the theory of finite element exterior calculus (FEEC). This talk will review the subject and especially recent progress connected to FEEC, which has led to very simple stable elements in two and three dimensions.