<p>We review our current knowledge of thermal and viscous instabilities in accretion discs around compact objects. We begin with classical disc models based on analytic viscosity prescriptions, discussing physical uncertainties and exploring time-dependent solutions of disc evolution. We also review the ionization instability responsible for outbursting dwarf nova and X-ray binary systems, including some detailed comparisons between alpha-based models and the observed characteristics of these systems. We then review modern theoretical work based on ideas around angular momentum transport mediated by magnetic fields, focusing in particular on knowledge gained through local and global computer simulations of MHD processes in discs. We discuss how magnetohydrodynamics (MHD) may alter our understanding of outbursts in white dwarf and X-ray binary systems. Finally, we turn to the putative thermal/viscous instabilities that were predicted to exist in the inner, radiation pressure-dominated regions of black hole and neutron star discs, in apparent contradiction to the observed stability of the high/soft state in black hole X-ray binaries.</p>