In this thesis we ~nvestigate three problems in the earth sciences where gravity currents play an important role. In the first part we consider two types of terrestrial gravity current: kata- . batic winds and submarine turbidity currents. We derive and solve the classical Prandtl model for katabatic wind flow in which the vertical wind profile is resolved. We show that this model breaks down when the slope becomes small, and pose a~ improved model which removes this singularity. Solutions of the improved model are compared with observations and output from a numerical model. We then investigate two layer-averaged models that are used to describe the flow of submarine turbidity currents. We find that both models predict that in some circumstances 'ignition' can occur, in which the current velocity becomes unbounded. We show that the only way this phenomenon can be prevented is by a decrease in the underlying slope. In the second part, we consider the unusual morphology of the Martian north polar ice cap. We use a model for the sublimation kinetics at the ice-atmosphere interface and include an explicit description of dust, both suspended in the atmosphere and frozen within the ice cap. Thansport of dust and ice are then included, and the model is investigated analytically and numerically. We find that this model can have multiple steady states, and that troughs may form during a transition between steady states. In this model, such a transition may be caused by obliquity-induced climate change.