  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.

Quantum Physics;Statistical Mechanics;Atomic Physics