One of the most important source of noise and vibrations associated with vehicles is the vibration of driveline systems. Such phenomena are subjectively associated with customer complaints. In this study the torsional vibrations of driveline systems were investigated using discretised and lumped mass models of the system. In the literature, many of the problems associated with torsional vibrations and refinement in drivelines have been tackled through relatively simple, lumped mass models combinedw ith experimentalm easurements. However, some problems remain particularly where instabilities occur or complex coupling with other vehicle vibration modes exists. The review of previous work showed that although it is important to understand the dynamic behaviour of the individual driveline components; for example, engine, clutch, gearbox, etc., the whole system must be analysed together because of all the coupling which occurs. The main source of excitation for torsional vibration of the driveline system is the engine fluctuating torque. A computer program using MATLAB subroutines was developed to obtain this fluctuation torque for different engine parameters for subsequent use in the modelling. A substructure approach, using stiffness coupling technique with combined use of residual flexibility and modal synthesis, was used to analyse free and forced vibrations of the system, as a linear system. A computer program using MATLAB subroutines was designed to facilitate application of this technique. Good agreement between results for the overall system model and substructure model was found even for a few considered modes. This substructure technique offers significant computational advantages over other methods. The effect of non-linear sources in the driveline system such as backlash, non-linear spring stifffiess, Hooke's joint and angularity of the propeller shaft on the system torsional vibrations was investigated. The effect of backlash in the driveline system was significant and, as expected, vibration levels increased as backlash increased. Hooke's joints caused an additional complex source of excitation but their significance is dictated by the details of the particular driveline design. The modelling showed that instabilities commonly referred to a shunt or shuffle could occur during clutch engagement. The stick slip frictional properties of the clutch were crucial in this behaviour and the relative importance of various design features was quantified. A mathematical model including torsional motion of the driveline system and other vehicle body motions was developed to analyse the ways in which the driveline couples with other dynamic components. Two running conditions were considered; steady state running and transient during clutch engagement. It was shown that the complete system was capable of self-excited oscillations under certain conditions during normal running as well as the instability which could occur during clutch engagement. This comprehensive model represents an important contribution of the work in this area of research in two ways. First, it clarifies understanding of the dynamic coupling between the rotational and translational components of the whole vehicle system. Second, it provides design information to tackle instability problems and to lead to reductions in overall vibration levels.