Control theory for mechanical systems is an area of active research in control theory. Such systems are characterised both by their diverse applications (e.g., robotics, vehicle locomotion, and space and flight dynamics), and by their rich mathematical structure. In this talk I shall focus on a certain mathematical structure which occurs in a large class of mechanical systems. These systems are characterised by the presence of an affine connection. The affine connection plays a beautiful role in the control theory for these systems, and we shall overview such topics as controllability (can you get from one state of the system to another) and optimal control (can you do it while minimising a cost function?)

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