On the elliptic modular function $j(\tau)$

In this mostly expository talk, I will discuss three aspects of the classical modular j-function: parabolic, elliptic, and hyperbolic. The first two are well-known theories of the monstrous moonshine and the complex multiplication. The third is rather new and is explained in a little more detail. My goal is to give the audience an impression that the three aspects are not independent, and there must be something still to be found on this very classical object.