The Shimura–Taniyama–Weil conjecture, now known as the modularity theorem, states that all elliptic curves over the field of rational numbers are modular. In this talk we will not attempt to discuss the proof; instead, we will have a more modest goal — to understand the statement, i.e., what it means for an elliptic curve to be modular. We will introduce elliptic curves, modular forms, Hecke algebras and eventually the Shimura construction. Time permitting, we will also see how the modularity theorem relates to other theorems and conjectures.