  Inversion and time reversal are essential symmetries for the structure of
Cooper pairs in superconductors. The loss of one or both leads to modifications
to this structure and can change the properties of the superconducting phases
in profound ways. Lacking inversion, superconductivity in noncentrosymmetric
materials has become an important topic, in particular, in the context of
topological superconductivity as well as unusual magnetic and magneto-electric
properties. Recently, crystal structures with local, but not global
inversion-symmetry breaking have attracted attention, as superconductivity can
exhibit phenomena not naively expected in centrosymmetric materials. After
introducing the concept of locally noncentrosymmetric crystals and different
material realizations, we discuss consequences of such local symmetry breaking
on the classification, the expected and, in parts, already observed
phenomenology of unconventional superconductivity, and possible topological
superconducting phases.
