The $j$-invariant is a special example of a modular form: a complex-valued function on the upper half-plane which behaves “nicely” respect to a group of symmetries. Modular forms have surprising connections with several areas of mathematics, in particular number theory. In this seminar we cover basic definitions of the theory of modular forms making emphasis on its connection with complex elliptic curves, from which the $j$-invariant arise naturally. Finally, we will try to illustrate how the $j$-invariant encodes meaningful information for several mathematical objects.