This book serves as an introduction to the use of mathematics in describing collective phenomena in physics and biology. Derived from a course of innovative lectures, the book shows students early in their studies how many of the topics they have encountered - partial differential equations, differential equations, Fourier series, and linear algebra - are useful in constructing, analysing and interpreting phenomena present in the real world. Throughout, ideas are developed using worked examples and exercises with solution. The text does not assume a strong background in physics.