  This is an introductory survey on cluster algebras and their (additive)
categorification using derived categories of Ginzburg algebras. After a gentle
introduction to cluster combinatorics, we review important examples of
coordinate rings admitting a cluster algebra structure. We then present the
general definition of a cluster algebra and describe the interplay between
cluster variables, coefficients, c-vectors and g-vectors. We show how c-vectors
appear in the study of quantum cluster algebras and their links to the quantum
dilogarithm. We then present the framework of additive categorification of
cluster algebras based on the notion of quiver with potential and on the
derived category of the associated Ginzburg algebra. We show how the
combinatorics introduced previously lift to the categorical level and how this
leads to proofs, for cluster algebras associated with quivers, of some of
Fomin-Zelevinsky's fundamental conjectures.
