Vibrations of a vehicle that moves on a long elastic structure can become unstable because of elastic waves that the vehicle generates in the structure. A typical example of the vehicle that can experience such instability is a high-speed train. Moving with a sufficiently high speed, this train could generate in the railway track elastic waves, whose reaction might destabilise vibrations of the train. Such instability could increase the level of vibrations of both the train and the railway track, significantly worsening the comfort of passengers and increasing the probability of the track deterioration and the train derailment. Instability of a moving vehicle on an elastic structure can be classified as one of the "moving load problems". This class of problems has been drawing attention of researches for more than a century being a fundamental issue in dynamics of bridges and railway tracks. Recently, the classical "moving load problem" has attracted researches once again because of the rapid development of high-speed railways. The necessity to take a fresh look at this old problem is based on the fact that in earlier studies it was usually assumed that the load speed is much smaller than the wave velocity in the elastic structure, which the load moves on. Nowadays, this assumption is no longer acceptable, since modern high-speed trains are able to move with a speed that is comparable with the wave velocity in a railway track. The main objective of this thesis is to study the stability of the train-track system at high speeds. The practical aim behind this objective is to develop an accurate and efficient method that would allow for choosing parameters of the train-track system so that the stability is guaranteed at operational train speeds. Having such a method developed, this thesis aims to study the effect of physical parameters of a moving train bogie on stability of the train-track system; analyse the effect of periodical inhomogeneity of the track that is caused by sleepers and rail corrugation on stability of the train-track system; investigate the effect of waves in the track subsoil on stability of the train-track system; To investigate the influence of the physical parameters of a vehicle, a simplified model for a railway track, namely a one-dimensional homogeneous elastically supported Timoshenko beam is used. Since instability depends on the reaction of elastic system, a so-called equivalent stiffness of the Timoshenko beam (a complex-valued function that depends on the frequency of vibrations of the contact point, its velocity and parameters of the beam and foundation) in a moving contact point is introduced and studied. For this development, the most important is the dependence of the equivalent stiffness on the velocity of the contact point. Therefore, this dependence is investigated thoroughly and then compared to that of an Euler-Bernoulli beam. Then, a two-mass oscillator, moving uniformly along such an elastic system is considered. It has been shown that vertical vibrations of this oscillator as it moves along the beam may become unstable if the oscillator's velocity exceeds the minimum phase velocity of waves in the beam. In this case, the equivalent dynamic stiffness of the beam has a negative imaginary part, which may be referred to as a "negative radiation damping" that is caused by radiation of anomalous Doppler waves. Instability domains in the parameter space of the system are found with the help of the D-decomposition method. The effect of various parameters of the system on its stability is studied. Then, a more realistic model for the vehicle is considered, namely a bogie that has two contact points with the beam. The bogie is modelled by a rigid bar of a finite length on two identical supports. The parametric analysis of the instability domain is performed with the emphasis on the effect of the bogie and the beam parameters and comparative analysis with simpler models (two-mass oscillator and simplified bogie) is carried out. Then, influence of the periodic inhomogeneity of elastic structure (actually, a railway track is periodically inhomogeneous due to the sleepers and also can become inhomogeneous due to corrugation of the rails) on the instability is studied. To this end, a simplistic model for the vehicle is utilised, namely a moving mass. The structure is modelled as an Euler-Bernoulli beam on visco-elastic foundation. The inhomogeneity is introduced by assuming that either the foundation stiffness or the beam cross-section is a periodic function of the co-ordinate. By moving on such a structure, the vehicle could experience parametric instability. It is found out that for high-speed trains, the zones of parametric instability are very narrow and, therefore should not be of concern. What could be a practically important threat is the instability that occurs when the minimum phase velocity of waves in the railway track is exceeded by the train. How large is this velocity? To answer this question, it is not enough to consider one-dimensional models of the railway track. The phase velocity of waves in a railway track is strongly influenced by the track subsoil. Therefore, to make a plausible estimation of train velocities at which the instability may arise, a three-dimensional model that includes the track subsoil should be employed. To this end, a railway track has been modelled by a beam resting on a visco-elastic half-space. The instability domain in the space of physical parameters of the system is found and parametrically studied with the help of the D-decomposition method. The main attention has been paid to the effect of the half-space parameters, especially to that of the material damping. It has been proved, that instability can occur at the velocities that are reachable for modern high-speed trains.