CUPID-0 is the first large mass experiment based on cryogenic calorimeters (bolometers) that implements the dual read-out of light and heat for background rejection. The detector, consisting of 24 enriched Zn$^{82}$Se crystals (5.28 kg of $^{82}$Se), is taking data in the underground LNGS (Italy) from March 2017. In this contribution we present the analysis that allowed to set the most stringent limit on the half-life of neutrino-less double beta decay of $^{82}$Se. We prove that the particle identification, enabled by the simultaneous read-out of heat and light, provides an unprecedented background level for cryogenic calorimeters of few 10$^{−3}$ counts/keV/kg/y. Finally, we discuss the impact of these results on next generation projects.