Highly supersonic, magnetized and differentially rotating, accretion flows provide an environment for the propagation of waves and the growth of instabilities unlike those normally encountered on Earth. In this dissertation, I investigate the properties of oscillations and instabilities in accretion discs, developing theoretical and numerical models to explore the physical nature of variability encountered in a variety of astrophysical contexts.

Through linear theory and semi-analytical calculations, I first consider the physical properties of magnetically altered inertial waves. ‘Trapped inertial waves’ provide an attractive explanation of the fast variability observed in the emission from low-mass black hole binary systems, but such oscillations can be affected by magnetic tension provided by large-scale poloidal magnetic fields threading the accretion disc. Through local and global analyses, I constrain the modification of trapped inertial waves by poloidal, toroidal and helical magnetic fields.

I then investigate the excitation of oscillations in deformed discs with eccentric, non-circular streamlines. Many processes can lead to the growth of eccentricity in accretion discs, and turbulence deriving from the excitation of inertial waves by a local parametric instability provides one mechanism for curbing this growth. However, eccentric discs also provide an environment for the excitation of additional, inherently global oscillations, and I present a framework facilitating a semi-analytical investigation of these modes.

I finally employ numerical simulations to explore the dynamics of accretion disc oscillations in the non-linear regime. I first follow the non-linear saturation of inertial waves driven by parametric resonance in non-relativistic discs, and confirm the growth of a second family of large-scale, low-frequency oscillations. Using a pseudo-Newtonian framework to approximate relativistic effects, I then demonstrate the excitation of trapped inertial waves through non-linear coupling with accretion disc deformations in a black hole accretion disc, providing preliminary evidence that trapped inertial waves can be excited even in the presence of MHD turbulence