This thesis explores aspects of the theory of accretion discs in two astrophysical environments; around young low-mass stars - T Tauri stars, and in mass transfer binaries. The importance of discs in our understanding of these systems is already well-established, and in this thesis I investigate extensions of the simplest theoretical disc models to incorporate the complexities suggested by recent observations. A particular aim is to consider the role that magnetic fields - including fields within the disc and those of the central star - may play in controlling the evolution of the star-disc system. The first part of the thesis investigates the effects of stellar magnetic fields on T Tauri accretion discs. There is abundant observational evidence, some of it summarised in the Introduction, that both discs and stellar magnetic activity are essential elements in the understanding of T Tauri stars, and the theoretical implications of this are developed. In Chapter 2 the main assumptions of the model are set out, and the steady-state properties of discs around magnetic T Tauri stars are derived. Chapter 3 develops a time-dependent disc model, which is used to examine the effect of time-varying magnetic fields on the disc. Variable stellar fields might occur if T Tauri stars harbour magnetic activity cycles analagous to those seen on the Sun, and I show that some of the long-term photometric variability in T Tauri systems could be caused by the influence of magnetic cycles on the accretion disc. Chapter 4 then follows the evolution of the star-disc system on much longer timescales by combining pre-main-sequence stellar evolution models with those for the disc. The resulting model is used to examine the rotation rates of magnetically braked T Tauri stars, and the possible influence of close binary companions on those rotation rates. The second part of the thesis considers accretion discs in interacting binary systems. In Chapter 5 I discuss some of the consequences of a magnetic dynamo operating within the disc, for the structure and evolution of discs in dwarf novae. A magnetic dynamo is a promising candidate mechanism for the origin of the viscosity in accretion discs, and this Chapter discusses the implications of an operating dynamo for observations of dwarf novae, which are among the best studied systems containing accretion discs. A simple model is presented in which the prominent outbursts seen in these systems have a direct origin in the physics of the underlying disc dynamo. Finally, Chapter 6 presents the results of three dimensional simulations of the interaction between the gas stream from the mass-donating star and the accretion disc. The hydrodynamic calculations show that a significant fraction of the stream gas can ricochet off the outer rim of the disc and overflow towards smaller radii. I discuss the implications of these results for models and observations of low-mass X-ray binaries and cataclysmic variables.