Heat transfer during melting and freezing in enclosures is important in the design of heat exchangers using phase change material (PCM) for thermal energy- storage. While freezing heat transfer is governed mainly by conduction, melting of the PCM generally invokes free convective currents in the melt phase depending on the orientation and geometry of the heat transfer surfaces. This paper deals with a computational investigation of the flow patterns and isotherms which develop during the melting of a PCM contained in a rectangular chamber. The heating surface(s) may be horizontal and/or vertical with the other sides insulated. Effects of heating both the vertical and horizontal faces as well as orientation of the cavity to the gravitational vector are examined. The shapes of the solid-liquid interface demonstrate clearly the generation of the cellular structure in the melt phase when the bottom horizontal face is heated. The cellular structure results in an undulating pattern for the heat transfer distribution on the bottom surface. The number of peaks and valleys in the Nusselt number distribution depends on the thickness of the melt layer.