The crystallization conjecture: a review

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 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.