Many deformable bodies are effectively incompressible. This means that the Jacobian determinants for all their deformations must be everywhere equal to 1. This talk will show by simple examples that the equations governing the motion of incompressible bodies are much more complicated than those governing the motion of compressible bodies (whose deformations need only preserve orientation), but have far more regularity. The role of incompressibility will be related to the question of constructing invariant dissipative mechanisms for hyperbolic conservation laws.