Models of quasi-periodic variability in neutron stars and black holes

Black holes and weakly-magnetic accreting neutron stars show strong quasi-periodic variability over timescales that span a very wide range. This property offer us a unique opportunity to improve our understanding of basic physical processes in strong gravitational fields as it reveals, for the first time, phenomena that occur on the most fundamental timescales near accreting compact objects. In this article, I review our current understanding of the variability properties of accretion flows into neutron stars and black holes. I discuss a number of challenges faced by theoretical models, in order to identify open questions in accretion physics that need to be addressed. Finally, I discuss the relation to and implications for variability models of the detection of nearly coherent oscillations during type I X-ray bursts in several neutron-star sources.