  Recent experimental breakthroughs in trapping, cooling and controlling
ultracold gases of polar molecules, magnetic and Rydberg atoms have paved the
way toward the investigation of highly tunable quantum systems, where
anisotropic, long-range dipolar interactions play a prominent role at the
many-body level. In this article we review recent theoretical studies
concerning the physics of such systems. Starting from a general discussion on
interaction design techniques and microscopic Hamiltonians, we provide a
summary of recent work focused on many-body properties of dipolar systems,
including: weakly interacting Bose gases, weakly interacting Fermi gases,
multilayer systems, strongly interacting dipolar gases and dipolar gases in 1D
and quasi-1D geometries. Within each of these topics, purely dipolar effects
and connections with experimental realizations are emphasized.
