  The aim of this paper is twofold. First, we set up the theory of elastic
matter sources within the framework of general relativity in a self-contained
manner. The discussion is primarily based on the presentation of Carter and
Quintana but also includes new methods and results as well as some
modifications that in our opinion make the theory more modern and transparent.
For instance, the equations of motion for the matter are shown to take a neat
form when expressed in terms of the relativistic Hadamard elasticity tensor.
Using this formulation we obtain simple formulae for the speeds of elastic wave
propagation along eigendirections of the pressure tensor. Secondly, we apply
the theory to static spherically symmetric configurations using a specific
equation of state and consider models either having an elastic crust or core.
