  This paper is based on a series of talks given at the ESI program on
'Mathematical Perspectives of Gravitation Beyond the Vacuum Regime' in February
2022. It is meant to be an introduction to the field of relativistic elasticity
for readers with a good base in the mathematics of General Relativity with no
necessary previous of knowledge of elasticity either in the classical or
relativistic domain. Despite its introductory purpose, the present work has new
material, in particular related to the formal structure of the theory.
