The present thesis is concerned with the mathematical modelling of a railway vehicle. The modelling does not only deal with the vehicle but also the track it runs on. Different models are described and investigated as to how they affect the dynamics of the vehicle. The bulk of the investigations is focussed on the stability of the vehicle. For this we introduce two stability criteria: the linear critical speed and the nonlinear critical speed. The linear critical speed is the vehicle speed at which the vehicle becomes unstable in a linear analyses, while the nonlinear critical speed is such that no oscillating solutions occur below this vehicle speed. The difference between a linear and a nonlinear analysis is hereby pointed out. The oscillating solutions found are analysed by applying methods from the nonlinear dynamics. By this periodic and chaotic solutions are described, for instance a scenario of a period adding sequence. For the vehicle we use a two-car test vehicle with a prototype of a single-axle bogie (a so-called KERF bogie). The vehicle is from the Danish State Railways and runs on the Copenhagen network. What is special about this vehicle is that the single axle bogie is steered by a mechanical steering system. Interest is focussed on the single-axle bogie. For simplification a model of the single-axle bogie alone is analysed under different modelling conditions. The dynamics of a model of the whole vehicle are investigated on: A) a straight track, B) a curved track, C) a track flexible in the vertical direction, D) a track with irregularities. Among other things, the investigations lead to the understanding of the influence of the stiffness in the steering system. On the irregular track the simulations are compared with corresponding measurements. Furthermore two different models are developed for the track: A simple model of the whole track as one rigid body following each wheelset. An elastic model where the rails are modelled as Euler-Bernoulli beams discretely supported by rigid sleepers. The simple model is used to find the influence of a flexible track on the dynamics of the single-axle bogie, while the elastic model is more useful to for example study the effect of changes in flexibility along the track. Finally measurements of the flexibility of tracks are described.