Let $X$ be a smooth, complex Fano $4$-fold, and $\rho(X)$ its Picard number. We will discuss the following theorem: if $\rho(X)>12$, then $X$ is a product of del Pezzo surfaces. This implies, in particular, that the maximal Picard number of a Fano $4$-fold is $18$. After an introduction and a discussion of examples, we explain some of the ideas and techniques involved in the proof.