Contact melting is the process during which a phase change material is placed in contact with a substrate that is at a temperature above the phase change temperature. This leads to melting of the phase change material and a °owing liquid layer which is being squeezed out due to the weight of the solid pushing down upon it. This process arises in many engineering problems such as the production of construction materials and alloys. Other uses include thermal storage systems that rely on the storage of energy as latent heat, which is released upon melting. The mathematical modelling of contact melting involves two heat equations, one in the solid and one in the liquid phase, coupled with the Navier-Stokes equations for the °ow in the melt, a Stefan condition at the phase change interface and a force balance between the weight of the solid and the countering pressure in the melt. Consequently, in this thesis we will deal with the one dimensional heat equation and move on to the problem commonly known as the Stefan problem - or moving boundary problem. We will consider both one and two phase problems and eventually look at a contact melting problem. The focus of this dissertation is to develop a three dimensional model to describe a contact melting process and to develop and apply an approximation method with minimal error.