We present a theoretical study of the onset of electric polarization close to a surface in magnetic materials and in thin films and multilayers. We consider two different paths that lead to the onset of multiferroic behavior at the boundary in materials that are bulk collinear ferromagnets or antiferromagnets. These two paths are distinguished by the presence or absence of a surface induced Dzyaloshinskii-Moriya interaction which can be taken into account through Lifshitz invariants in the free energy of the system. Experimental consequences are discussed in the light of the theory.