  A quiver is an oriented graph. Quiver mutation is an elementary operation on
quivers. It appeared in physics in Seiberg duality in the nineties and in
mathematics in the definition of cluster algebras by Fomin-Zelevinsky in 2002.
We show, for large classes of quivers Q, using quiver mutation and quantum
dilogarithms, one can construct the combinatorial DT-invariant, a formal power
series intrinsically associated with Q. When defined, it coincides with the
"total" Donaldson-Thomas invariant of Q (with a generic potential) provided by
algebraic geometry (work of Joyce, Kontsevich-Soibelman, Szendroi and many
others). We illustrate combinatorial DT-invariants on many examples and point
out their links to quantum cluster algebras and to (infinite) generalized
associahedra.
