The Langlands program connecting Galois representations and automorphic forms is one of the ambitious visions in modern number theory. Its local version predicts a relation between local Galois representations and representations of reductive groups over local fields. In what is called the \emph{classic} case, the representations are smooth with coefficients in a field with characteristic zero; here the local Langlands correspondence is proved (Harris-Taylor, Henniart).

In 2000, a $p$-adic component of this program was conjectured by Breuil and Mézard. The $p$-adic Hodge theory originated by Fontaine settles the Galois side; on the other side, however, the theory of $p$-adic representations of $p$-adic reductive groups remains a burning question. Answers have been given mostly about the case of ${\rm GL}_2(\qpp)$ for which a local Langlands correspondence has been formulated and proved by Colmez in 2007.

By mod $p$ reduction, this theory leads to the one of the smooth representations of $p$-adic reductive groups over an algebraic closure $\fp$ of $\fpp$. It is a singular point in the framework of representations of $p$-adic groups because most of the basic methods suitable for the classic case prove obsolete.

I will give an overview of (some of) the methods used to tackle the mod $p$ representations of ${\rm GL}_n(F)$ where $F$ is a $p$-adic field. By describing what is now known about them, I will provide an idea of the stakes of a potential mod $p$ Langlands correspondence for ${\rm GL}_n(F)$.