An introduction to the mathematical fluid dynamics of oceanic and atmospheric flows The study of the movement of the atmosphere and the oceans is intriguing challenging and important particularly in the context of current concerns about the climate. The familiar and tested approach to these problems is based on the construction of model equations tailored to address specific flow scenarios. In this text we present a single over arching approach which uses the thin shell approximation – and nothing more – applied to the general equations of fluid dynamics. This allows a range of classical problems and some new applications to be accessed from a single formulation which retains all the relevant physical attributes as well as the essential characteristics of the spherical geometry. The approximations and assumptions are clear and higher order terms are readily accessible. The main aim is to present the material in a mathematically consistent and robust fashion – in the applied sense – emphasising the systematic asymptotic methods usually employed in mathematical fluid dynamics. This is not a textbook that introduces the physical principles underpinning the study of the oceans and the atmosphere. Rather it is intended to enhance the more usual modelling approach to these studies and more significantly to introduce those with mathematical interests but no expertise in these particular applications to these types of problems. The text is suitable for researchers and students in the oceanic and atmospheric sciences and for mathematicians – again researchers and students – with an interest in the application of fluid dynamics to more complicated flow scenarios.