  We review the recent developments and the current status in the field of
quantum-gas cavity QED. Since the first experimental demonstration of atomic
self-ordering in a system composed of a Bose-Einstein condensate coupled to a
quantized electromagnetic mode of a high-$Q$ optical cavity, the field has
rapidly evolved over the past decade. The composite quantum-gas--cavity systems
offer the opportunity to implement, simulate, and experimentally test
fundamental solid-state Hamiltonians, as well as to realize non-equilibrium
many-body phenomena beyond conventional condensed-matter scenarios. This hinges
on the unique possibility to design and control in open quantum environments
photon-induced tunable-range interaction potentials for the atoms using
tailored pump lasers and dynamic cavity fields. Notable examples range from
Hubbard-like models with long-range interactions exhibiting a
lattice-supersolid phase, over emergent magnetic orderings and quasicrystalline
symmetries, to the appearance of dynamic gauge potentials and non-equilibrium
topological phases. Experiments have managed to load spin-polarized as well as
spinful quantum gases into various cavity geometries and engineer versatile
tunable-range atomic interactions. This led to the experimental observation of
spontaneous discrete and continuous symmetry breaking with the appearance of
soft-modes as well as supersolidity, density and spin self-ordering, dynamic
spin-orbit coupling, and non-equilibrium dynamical self-ordered phases among
others. In addition, quantum-gas--cavity setups offer new platforms for
quantum-enhanced measurements. In this review, starting from an introduction to
basic models, we pedagogically summarize a broad range of theoretical
developments and put them in perspective with the current and near future
state-of-art experiments.
