  We review the derivative expansion (DE) method in Casimir physics, an
approach which extends the proximity force approximation (PFA). After
introducing and motivating the DE in contexts other than the Casimir effect, we
present different examples which correspond to that realm. We focus on
different particular geometries, boundary conditions, types of fields, and
quantum and thermal fluctuations. Besides providing various examples where the
method can be applied, we discuss a concrete example for which the DE cannot be
applied; namely, the case of perfect Neumann conditions in 2 + 1 dimensions. By
the same example, we show how a more realistic type of boundary condition
circumvents the problem. We also comment on the application of the DE to the
Casimir-Polder interaction which provides a broader perspective on
particle-surface interactions.
