  Many of the properties of superconductors related to quantum coherence are
revealed when the superconducting state is forced to vary in space - in
response to an external magnetic field, a proximity contact, an interface to a
ferromagnet, or to impurities embedded in the superconductor. Among the
earliest examples is Andreev reflection of an electron into a retro-reflected
hole at a normal-superconducting interface. In regions of strong inhomogeneity
multiple Andreev reflection leads to the formation of sub-gap states, Andreev
bound states, with excitation energies below the superconducting gap. These
states play a central role in our understanding of inhomogeneous
superconductors. The discoveries of unconventional superconductivity in many
classes of materials, advances in fabrication of superconducting/ferromagnetic
hybrids and nano-structures for confining superfluid $^3$He, combined with
theoretical developments in topological quantum matter have dramatically
expanded the significance of branch conversion scattering and Andreev bound
state formation. This collection of articles highlights developments in
inhomogeneous superconductivity, unconventional superconductivity and
topological phases of superfluid $^3$He, in which Andreev scattering and bound
states underpin much of the physics of these systems. This article provides an
introduction to the basic physics of Andreev scattering, bound-state formation
and their signatures. The goal is both an introduction for interested readers
who are not already experts in the field, as well as to highlight several
examples in which branch conversion scattering and Andreev bound states provide
unique signatures in the transport properties of inhomogeneous, topological and
unconventional superconductors.
