An elliptic curve is a curve of genus 1 together with a point on it. The theory of elliptic curves is at the intersection of arithmetic and complex geometry, as they are essentially the same as smooth plane cubics and complex tori of dimension 1. The main aim of this talk is to give an introduction to this subject. In the last part of the talk I will discuss the j-invariant, which represents the first step for the study of moduli of elliptic curves.