The talk will review the Virtual Element spaces, recently introduced by the speaker and co-authors in order to approximate spaces of the type H(div) or H(curl) on polygonal and polyhedral decompositions. As is well known, these spaces are needed in the approximation of boundary value problems for (systems of) Partial Differential Equations in mixed form. The Virtual Element spaces are defined locally as solutions of systems of PDE's, that however don't need to be solved (not even in an approximate way).