This free course contains an introduction to metric spaces and continuity. The key idea is to use three particular properties of the Euclidean distance as the basis for defining what is meant by a general distance function, a metric. Section 1 introduces the idea of a metric space and shows how this concept allows us to generalise the notion of continuity. Section 2 develops the idea of sequences and convergence in metric spaces. Section 3 builds on the ideas from the first two sections to formulate a definition of continuity for functions between metric spaces.