Optimisation can be thought of as the process of minimising costs or maximising benefits, or finding the best possible compromise between the two. It is an important tool in decision-making, and there is even a theory that the structures of animals, their movement and behaviour, have been influenced by optimisation through evolution. In this course we develop some of the mathematics that deals with different aspects of optimisation, with and without constraints. Specific topics include: maxima and minima of functions of several variables, linear programming, the calculus of variations, network flows, transportation and allocation problems.