Symmetric differentials are sections of the symmetric powers of the sheaf of differentials. Geometrically they describe multi-foliations, in the same way differential 1-forms (i.e symmetric differential of order 1) describe foliations. Information on the existence and asymptotic growth of symmetric differentials gives important information on the geometry of the projective variety, e.g. hyperbolicity. We discuss the non-existence of symmetric differentials on subvarieties of $ P^n$ of low codimension, the jumping phenomenon and applications toward the hyperbolicity of hypersurfaces of $ P^3$. The jumping phenomenon refers to the number of symmetric differentials along a family of projective varieties. This contrasts with the constancy of the plurigenera (the dimension of the space of sections of powers of the canonical bundle).