Large amplitude interfacial waves are an important feature of annular gas-liquid two-phase flow. They act as a source of entrainment for liquid droplets. They occur for liquid flow rates above a critical value which depends on the gas flow rate. This thesis examines the formulation of a mathematical model to describe the behaviour of these nonlinear waves. Attention is focussed on the case of vertical upwards flow with reference to the experimental conditions for the rig at AERE Harwell. A comprehensive account is given of the limitations and similarities of mathematical models proposed by earlier research workers and their applicability to vertical two-phase flow. The most suitable approaches are found to be kinematic wave theory and an integral method. Experiments have been carried out at AERE Harwell to determine the relationship between liquid flux and film thickness required by kinematic wave theory and also to test some of the theory's predictions. There is a discussion of the difficulties involved in modelling the stresses exerted by the turbulent gas core on disturbance waves. The applicability of Benjamin's 'quasi-laminar' theory is considered. A linear stability analysis indicates that the interface is always unstable. The linear theory cannot provide a criterion for disturbance wave inception. Alternative explanations for wave inception are suggested. The SMAC (Simplified Marker And Cell) numerical method has been developed to model the time dependent behaviour of large amplitude waves in vertical annular two-phase flow. Finally, it is proposed that any realistic mathematical model for disturbance waves should be based upon kinematic wave theory and should take account of wave-breaking.