The aim of this research project was to develop a discontinuous Galerkin method for two-fluid flows, which is accurate, versatile and can alleviate some of the problems commonly encountered with existing methods. A novel numerical method for two-fluid flow computations is presented, which combines the space-time discontinuous Galerkin finite element dis- cretization with the level set method and cut-cell based interface tracking. The space-time discontinuous Galerkin (STDG) finite element method of- fers high accuracy, an inherent ability to handle discontinuities and a very local stencil, making it relatively easy to combine with local hp-refinement. A front tracking approach is chosen because these methods ensure a sharp interface between the fluids are capable of high accuracy. The front tracking is incorporated by means of cut-cell mesh refinement, because this type of refinement is very local in nature and hence combines well with the STGD. To compute the interface dynamics the level set method (LSM) is chosen, because of its ability to deal with merging and breakup, since it was ex- pected that the LSM combines well with the cut-cell mesh refinement and also because the LSM is easy to extend to higher dimensions. The small cell problem caused by the cut-cell refinement is solved by using a merg- ing procedure involving bounding box elements, which improves stability and performance of the method. The interface conditions are incorporated in the numerical flux at the interface and the STDG discretization ensures that the scheme is conservative as long as the numerical fluxes are conserva- tive. All possible cuts the 0-level set can make with square and cube shaped background elements are identified and for each cut an element refinement is defined explicitly. To investigate the numerical properties and performance of the numeri- cal algorithm it is applied to a number of one and two dimensional single and two-fluid test problems. Also, the Object Oriented Programming (OOP) design and implementation of the two-fluid method were discussed.