The aim of this course is to give an introduction to abelian varieties over the complex numbers, starting from dimension 1, that is, complex elliptic curves, which we will present as motivation for the course. The main object we will work with is the complex torus, that is, the quotient of a complex vector space by a full-rank lattice. We will see that complex abelian varieties are complex tori, and characterize when a complex torus is an abelian variety. As an example, we will introduce the Jacobian of a curve, see how to construct it as a complex abelian variety, and define Riemann theta functions and their role in relating a curve with its complex Jacobian. As an application, we will venture into the theory of complex multiplication: we will see how one can construct abelian varieties with this property, and how to use this construction together with Riemann theta functions in order to obtain equations of curves with interesting properties.