This work analyses models for two-phase flows and methods for the numerical resolution of these models. It is therefore one contribution to the development of reliable design tools for multiphase applications. Such tools are needed and expected by engineers in a range of fields. The approximate Riemann solver of Roe is studied. Roe schemes for three different two-phase flow models are implemented. The schemes are analysed by calculation of benchmark tests, and by comparison with each other. An isentropic discrete-equation multiphase model is presented. The incorporation of different interfacial-pressure models is discussed, and examples are given. The purpose of the multi-stage (MUSTA) method is to come close to the accuracy of upwind schemes while retaining the simplicity of centred schemes. Here it is applied to the drift-flux model. The main advantage of the MUSTA scheme is its simplicity. A multiphase characteristic-based boundary-condition method is been tested, and it is shown to be workable for transient problems. The intended audience is researchers, graduate students and engineers working with two-phase flow modelling and numerical methods.