In this talk, we would focus on Sobolev spaces in the context of models of Nonlinear Elastcity. Since we want homeomorphisms to be in the class of admissible mappings for such models, they are thoroughly studied (as well as their strong or weak closures in the given Sobolev space). Several nice properties can be studied, e.g. the sign of the Jacobian (representing that the deformation is sense-preserving), different notions of injectivity (representing that interpenetration of matter does not occur) or whether the mapping can be approximated by more regular ones. This talk should be accessible also for mathematicians working in fields outside of Calculus of Variations and Nonlinear Elasticity.