  Compass models are theories of matter in which the couplings between the
internal spin (or other relevant field) components are inherently spatially
(typically, direction) dependent. Compass-type interactions appear in diverse
physical systems including Mott insulators with orbital degrees of freedom
(where interactions sensitively depend on the spatial orientation of the
orbitals involved), the low energy effective theories of frustrated quantum
magnets, systems with strong spin-orbit couplings (such as the iridates),
vacancy centers, and cold atomic gases. Kitaev's models, in particular the
compass variant on the honeycomb lattice, realize basic notions of topological
quantum computing. The fundamental inter-dependence between internal (spin,
orbital, or other) and external (i.e., spatial) degrees of freedom which
underlies compass models generally leads to very rich behaviors including the
frustration of (semi-)classical ordered states on non-frustrated lattices and
to enhanced quantum effects prompting, in certain cases, the appearance of zero
temperature quantum spin liquids. As a consequence of these frustrations, new
types of symmetries and their associated degeneracies may appear. These
intermediate symmetries lie midway between the extremes of global symmetries
and local gauge symmetries and lead to effective dimensional reductions. We
review compass models in a unified manner, paying close attention to exact
consequences of these symmetries, and to thermal and quantum fluctuations that
stabilize orders via order out of disorder effects. We review non-trivial
statistics and the appearance of topological quantum orders in compass systems
in which, by virtue of their intermediate symmetry standard orders do not
arise. This is complemented by a survey of numerical results. Where appropriate
theoretical and experimental results are compared.
