The microwave heating of a one-dimensional, semi-infinite material with temperature-dependent material properties is studied. The governing equations, Maxwell's equations and the forced heat equation, are considered in the limit of high-frequency radiation and small thermal diffusivity. Particular power-law dependencies are chosen for the material parameters and an uniformly valid asymptotic solution is derived, using the techniques of strained coordinates and multiple scales. The asymptotic solution is compared with numerical solutions and an example involving the development of a hot-spot (a localised area of very high temperature) is examined. © 1994.