instance [@FK09], [@DWZ09], [@GLS10], [@CKLP13], ... . On the other hand, it has sparked many developments in representation theory, as illustrated by the introduction of the theory of $\tau$-tilting in [@AIR14], the study cluster-tilted algebras and their representations initiated in [@BMR07] and the study of representations of certain quivers with loops in [@GLS14], among other examples.

Central in the study of this link are cluster characters. Broadly speaking, they are maps which associate to each module over certain algebras (or object in certain triangulated categories) an element in a certain cluster algebra. They have been introduced in [@CC06], and have been studied, used and generalized for instance in [@CK08], [@CK06], [@FK09], [@Palu08], [@Plamondon09], [@Rupel15], ... .

The aim of these notes is to introduce cluster characters, present some of their main properties, and show how they can be used to categorify cluster algebras.

The notes are organized as follows. In Section \[sect::representations\], we introduce $F$-polynomials of modules over finite-dimensional algebras. They can be seen as a “homology-free” version of cluster characters. Their definition relies heavily on representation theory of quivers and on projective varieties called submodule Grassmannians; these are introduced first.

In Section \[sect::clustercat\], we introduce the cluster category of 