This paper discusses the application of an r-refinement, moving mesh technique for the solution of heat transfer problems with natural convection and phase change. The moving mesh technique keeps the number of elements and their connectivity fixed and clusters the nodes towards the phase change front at the expense of the solution of an extra differential equation. The governing differential equations describing the physical problem are modified to account for the mesh movement between time steps. The energy conservation equation uses the apparent heat capacity method to take into account the latent heat of phase change. The finite element discretization of all equations is presented. Several test problems are solved and the moving mesh FEM results are in a very good agreement with those in the published literature. The sensitivity of the results to variations of some user-definable computational parameters is found to be low, which means that the moving mesh method may be used without extensive previous experience. Its basic advantage is that less elements may be used to achieve accurate results.