Modular forms and functions can be defined as complex analytic functions on the upper half plane obeying a certain functional equation with respect to a certain discrete group action. From another perspective, they correspond to fairly natural forms and functions on spaces parametrizing elliptic curves, possibly with additional structure. As such, they played an important role in the development of the classical theory of abelian functions in mathematics (and physics) and for solutions of several problems in number theory. On the other hand, the subject became extremely popular when the connection of the Taniyama-Shimura-Weil conjecture to Fermat’s last theorem was discovered. Fermat’s last theorem was proven in 1994 by Wiles by resolving a special case of the conjecture. The full conjecture was proven in 2001 by Breuil, Conrad, Diamond and Taylor. The course aims to explain the significance of modular forms and the story of the conjecture, however not its proof.