This PhD work is about limitations of high intensity proton beams observed in the CERN Proton Synchrotron (PS) and, in particular, about issues at injection and transition energies. With its 53 years, the CERN PS would have to operate beyond the limit of its performance to match the future requirements. Beam instabilities driven by transverse impedance and aperture restrictions are important issues for the operation and for the High-Luminosity LHC upgrade which foresees an intensity increase delivered by the injectors. The main subject of the thesis concerns the study of a fast transverse instability occurring at transition energy. The proton beams crossing this energy range are particularly sensitive to wake forces because of the slow synchrotron motion. This instability can cause a strong vertical emittance blow-up and severe losses in less than a synchrotron period. Experimental observations show that the particles at the peak density of the beam longitudinal distribution oscillate in the vertical plane due to a short range wake field and following a travelling wave of about 700 MHz. In order to perform measurements, a dedicated single bunch beam was set up with a zero chromaticity plateau around transition energy. Extensive measurements were performed of the dynamics of instability in order to compute rise times and intensity thresholds. These measurements were done for several peak densities and the results show that the longer the bunch length, the higher is the threshold in intensity. Other measurements performed with a small negative chromaticity and another working point show that the intensity threshold can be pushed at higher values—in this case, the threshold was increased by 20%. The particularity of this work is that the instability is triggered during the acceleration. At transition energy, the momentum compaction factor is zero and the exchange of particles between the head and the tail of the beam from synchrotron motion, which is a natural way to damp instabilities, vanishes for few turns. Therefore the measurements at which η the instability is triggered are fundamental to know in which longitudinal regime the instability develops. Macro particle simulations were performed in order reproduce the dynamics of the instability with the HEADTAIL code, that simulates the beam interaction with the impedance of the machine. In the case of the PS, a very detailed impedance model does not exist, therefore a very simple resonator impedance was considered. The parameters of the code were adapted in order to simulate the beam as close as possible to the experimental conditions. The simulations showed that the travelling wave is well reproduced and therefore rise times were extracted for different beam intensities and compared to the measurements. A good agreement is found for a such simple impedance model. The intensity thresholds are reproduced for zero chromaticity within 30% and it is clear that some damping mechanisms occurring in the measurements are not reproduced by the code. In particular, octupolar components of the field, non-linear coupling and transverse space charge are not taken into account. These limitations could cause some discrepancies between model and measurements. This study allows to establish a broadband impedance model which can be use to predict the dynamics of the instability. A few experiments were done also with the use a gamma transition jump. In this case, the instability appears in the adiabatic regime and the rising of the instability is following the peak density of the beam. Both the intensity threshold measurements and the simulations show that the gamma jump is by far the most efficient way to increase significantly the intensity threshold. The choice of adequate working point appears also to be a cure of the instability. This study with the support of measurements provides a rough impedance model confirmed by simulations and an understanding of the different mechanisms triggering the instability. The last part of the thesis is dedicated to proton beam losses studies at injection energy. An eventual upgrade of the high intensity beam is limited by large beam losses at injection in the PS. Losses were measured with the current beam loss monitor system of the PS that they occur while the incoming beam enters in the machine and then during several hundred turns while the beam is circulating. Optics measurements were performed to check the mismatch of the transfer line between the PSBooster and the PS. A significant horizontal dispersion mismatch was found and in particular a difference in optics between the four injection lines of the PS. Aperture bottlenecks were found at the injection septum in both transverse planes and at the maximum of the injection bump. With the help of Monte Carlo simulations, the high radiation outside the ring induced by the losses was understood. However, the mismatch does not explain the continuous losses after the beam is injected. A possible explanation is that the space charge forces contribute to make particles cross resonances and are lost while they are oscillating at high amplitudes. The transverse phase space is refilled due to the combination of the synchrotron motion and space charge. Several solutions were proposed to improve the situation. A new optics of the transfer line and at injection could make the beam size smaller. Some special quadrupoles already installed in the PS could be used to adapt the optics of the incoming beam. Increase the injection energy from 1.4 GeV to 2 GeV kinetic would cause a gain of 65% in space charge forces, and these studies are currently ongoing for the upgrade of the PS in the framework of high-intensity LHC upgrade project. High intensity beams are also perturbed by space charge forces and transverse instabilities. Coherent tune shift measurements were performed at injection and extraction in order to evaluate the transverse imaginary part of the impedance and the contribution of the space charge to betatron frequency shift with the beam intensity.