In this thesis, a computational approach is used to study two-phase flow including phase change by direct numerical simulation. This approach follows the interface with an adaptive moving mesh. The incompressible Navier-Stokes equations are solved, in two-dimensional and axisymmetric geometries, using the Finite Element Method (FEM). The computational domain is discretized using an unstructured triangular mesh and the mini element is used to satisfy the "inf-sup" compatibility condition. A combination of smoothing mesh velocities and remeshing is used to preserve the mesh quality. Adaptive mesh refinement is used to keep the mesh sufficiently refined where needed. Mesh adaptation strategies, allowing to control the refinement of the computational mesh, are discussed in the context of specific applications. The accurate representation of the interface between the phases is a key issue to model surface tension dominated flows. Here, the interface is represented explicitly by nodes and segments that are a subset of the computational mesh and a sharp transition of the fluid properties can be achieved. The surface tension force is included as a singular volume force, like in the continuum model (CSF). The present discretization is shown to allow for exact equilibrium (up to rounding errors) between the pressure and surface tension terms. This is important in order to suppress spurious currents, which are a common issue in computational two-phase flow. However, an exact computation of the interface curvature is necessary for the spurious currents to be numerically zero. The curvature of the interface, is efficiently and accurately computed by using the Frenet-Serret formulas. A phase change model is implemented via a source term in the continuity equation, which is computed from the jump in conductive heat flux at the interface. The presented approach is shown to provide an accurate description of different two-phase flow phenomena, including phase change, and to handle cases with large material property ratios. Accuracy and robustness of the present method are demonstrated on several benchmark cases, where the results are compared to analytical or semi-analytical solutions and experimental data.