Critical hunting speed and running safety are important aspects of the dynamic behaviour of railway vehicles. In this paper, a study of the dynamic behaviour of a single railway vehicle truck running on tangent and curved tracks is carried out using a mathematical model of the combined system. It is seen from literature that there are no reported studies using the combined truck/curved track system in which rail lateral displacement is computed considering two point wheel/rail contact for evaluation of critical hunting speed, as well as derailment due to wheel climb. In this study, the truck consists of the truck frame with suspended wheelsets, while the track is idealized as a laterally flexible rail modelled as a spring mass damper system. The combined truck/track system has 10 degrees of freedom (DOF), consisting of the lateral displacement and yaw angle of the wheelsets and truck frame, as well as the lateral displacement of the front and the rear left and right rails. Equations of motion using a model with single-point and two-point wheel-rail contact are derived. Non-linearities in the wheelset model include the non-linear wheel-rail profile and the friction-creep characteristics of the wheel-rail contact geometry. The longitudinal and lateral primary and secondary suspensions are assumed to have linear stiffness and damping characteristics. A combination of linear Kalker's theory and non-linear heuristic creep model is adopted to calculate the creep forces. The mathematical equations of motion are solved using fourth order Runge-Kutta method, which requires that the second order differential equations be transformed into a set of first order differential equations. The transformed state space equations are solved in the time domain to obtain the dynamic response of a conventional truck, moving on tracks of various radii. The numerical simulation is done using MATLAB.