Optimization problems, in which one wants to find the values of variables to maximize or minimize an objective function subject to constraints on which variables are allowed, are common throughout the physical and biological sciences, economics, finance and engineering. This course looks at the formulation of optimization problems as mathematical problems, characterising solutions using necessary and/or sufficient optimality conditions and modern numerical methods and software for solving the problems. Both finite dimensional problems which involve a vector of variables, including linear and nonlinear programming, and infinite dimensional problems where the variables are functions, including optimal control problems, are covered.