Accretion discs are fluid-dynamical entities which surround many black holes. Observations reveal that these systems exhibit variability on a range of time scales. This thesis investigates phenomena occurring in black-hole accretion discs which are likely to induce high-frequency quasi-periodic variability. Two classes of pseudo-relativistic theoretical models are investigated. The first is based on the stability of transonic accretion flows and its connection to a disc instability that takes the form of propagating waves (viscous overstability). The time-dependent study looks at the conditions under which the transition between subsonic disc-like accretion, which occurs at large radii, and the supersonic flow characteristic of the immediate vicinity of the black hole is stable. In agreement with previous findings, results indicate that the system reaches a steady state for low viscosity. Above that threshold the transonic solutions are unstable to viscous overstability. The overstable inertial-acoustic waves appear to be excited near the maximum of the epicyclic frequency and are global in the sense that their frequency is maintained for a wide range of radii. The second class of models looks at accretion-disc oscillations which are trapped due to the non-monotonic variation of the epicyclic frequency in relativistic flows. In particular, it focuses on inertial waves trapped below the maximum of the epicyclic frequency which are excited in deformed, warped or eccentric, discs. The excitation mechanism involves a non-linear coupling between the global deformation, an intermediate wave and the inertial mode and results, under a variety of conditions, in growth of the latter. Excitation is only effective when global deformations are capable of reaching the inner disc with non-negligible amplitude. With that in mind, the conditions favourable to the propagation of warped and eccentric modes from the outer to the inner regions are analysed. Another aspect that is taken into account is the influence of a transonic background, ignored in the coupling calculations, on the propagation of modes in the disc. It is found that, under certain conditions, inertial waves may be severely affected or destroyed in this background. On the other hand, results indicate that the decay rate of inertial waves due to the presence of the radial inflow is small in sufficiently thin discs. In this case, the coupling mechanism can still work to excite trapped inertial modes.