The classical picture of an accretion disc is of a geometrically thin, nearly axisymmetric, fluid flow undergoing supersonic circular motion and slowly accreting due to angular momentum transport by the disc turbulence. There are, however, strong theoretical and observational grounds for considering non-circular motion in accretion discs. In this thesis I study the dynamics of such non-circular accretion discs using a mix of analytical and semi-analytical methods. One such example of a non-circular accretion disc is an eccentric disc, where the dominant fluid motion consists of slowly evolving, nested, confocal, Keplerian ellipses. I focus on the dynamics of these eccentric discs, along with the dynamical non-axisymmetric vertical structure that they set up.

In the first part of this thesis I consider eccentric waves in ideal fluid discs. I present a secular Hamiltonian theory describing the evolution of the disc orbits due to pressure gradients and show that it can be used to calculate the eccentric standing wave patterns in the disc (the eccentric modes). I derive a ``short wavelength'' theory for nearly circular orbits that nevertheless can have substantially non-axisymmetric surface density/pressure due to nonlinear eccentricity gradients and disc twist. I use this to show that the pressure profile and precessional forces (such as general relativistic apsidal precession) can focus eccentric waves, causing them to become highly nonlinear in the inner disc.

In the second part of this thesis I consider the action of non-ideal terms on the eccentric disc, such as excitation and damping by viscosity. I derive the ordinary differential equations describing simple horizontally invariant ``laminar flows'' in a local model of an eccentric disc, which can be used to study the disc's dynamical vertical structure. I also move beyond the purely hydrodynamic models of the preceding chapters and consider the action of magnetic fields in a local model.

Finally I apply the theory presented in this thesis to the highly eccentric discs expected from the tidal disruption of a star by a supermassive black hole. It is currently an open question how, and indeed if, the discs in tidal disruption events circularise. As a step towards understanding the evolution of the disc orbits, I calculate the dynamical vertical structure of highly eccentric discs, emphasising the role of radiation pressure and thermal stability, and showing that magnetic fields may be important where the disc is highly compressed near the periapsis.