Analytic solutions for heat transfer problems are limited to simple geometries and boundary conditions. This thesis aims to investigate numerical methods for solving transient heat transfer problems, with particular emphasis on problems involving localized heating that can lead to phase change. Such problems are of increasing importance in modelling additive manufacturing processes. Heat transfer is present in most engineering applications. Complex problems require numerical solutions which involve complex geometries, non-uniform boundary conditions and transient temperature history. For such problems, finite difference and finite element solution techniques have been developed. This thesis introduces the basic numerical models for finite difference and finite element calculations. Then the found calculation problems are employed to solve validation problems and generic heat transfer problems. A comparison is drawn to commercial code, namely Abaqus. Finally, melting and freezing processes are modelled and evaluated.