In nonlinear elasticity theory there is a class of models that can be derived from an action functional with the potential energy described by a strain energy density function. These are the hyperelastic materials. The simplest example is the Neo-Hookean material (like some plastics, biological tissues, or rubber). In this talk, after describing the equations of motion of an incompressible neo-Hookean material, I will concentrate on the Cauchy problem for those equations. There are similarities with the Euler equations of fluid dynamics, and there are major differences. I will show some new analytical tools that help with the low regularity well-posedness results. Based on joint work with Lars Andersson.