In this chapter we introduce the important notion of an ‘elliptic curve’. Elliptic curves are a natural class of plane curves that generalise the straight lines and conic sections studied in nearly all university mathematics courses (and many high school courses). However, the study of elliptic curves involves two new ingredients. First, it is useful to consider complex curves, not just real ones. Second, for some purposes it is more satisfactory to work in complex projective space rather than the complex plane C2. (Algebraic geometers call C the complex line because it is 1-dimensional over C. So C2 becomes the complex plane.) We introduce these refinements in simple stages.