  In this article we describe the crystallization conjecture. It states that,
in appropriate physical conditions, interacting particles always place
themselves into periodic configurations, breaking thereby the natural
translation-invariance of the system. This famous problem is still largely
open. Mathematically, it amounts to studying the minima of a real-valued
function defined on $\mathbb{R}^{3N}$ where $N$ is the number of particles,
which tends to infinity. We review the existing literature and mention several
related open problems, of which many have not been thoroughly studied.
