In the last twenty years, Rydberg atoms have become a versatile and much studied system for implementing quantum many-body systems in the framework of quantum computation and quantum simulation. However, even in the absence of coherent evolution Rydberg systems exhibit interesting and non-trivial many-body phenomena such as kinetic constraints and non-equilibrium phase transitions that are relevant in a number of research fields. Here we review our recent work on such systems, where dissipation leads to incoherent dynamics and also to population decay. We show that those two effects, together with the strong interactions between Rydberg atoms, give rise to a number of intriguing phenomena that make cold Rydberg atoms an attractive test-bed for classical many-body processes and quantum generalizations thereof
