Heat transfer with change of phase (freezing or melting) is important in numerous scientific and engineering applications. Since the pioneering works of Neumann and Stefan, a number of analytical and numerical techniques have been developed to deal with freezing and melting problems. One such analytical tool is the method of perturbation expansions, which is the main focus of this work. The article begins with a review of the perturbation theory and outlines the regular perturbation method, the method of strained coordinates, the method of matched asymptotic expansions, and the recently developed method of extended perturbation series. Next, the applications of these techniques to phase change problems in Cartesian, cylindrical, and spherical systems are discussed in detail. Although the bulk of the discussion is confined to one-dimensional situations, the report also includes two- and three-dimensional cases where admittedly the success of these techniques has so far been limited. The presentation is sufficiently detailed so that even the reader who is unfamilar with the perturbation theory can understand the material without much difficulty. However, at the same time, the discussion covers the latest literature on the subject and therefore should serve as the state-of-the-art review.