Cluster algebras, quiver mutation and triangulations

Cluster algebras have been introduced a decade ago by S. Fomin and A.
Zelevinsky. The original goal was to provide an algebraic framework for
phenomena occuring in dual canonical bases of (quantized) universal
enveloping algebras and of total positivity in algebraic groups in order
to get a better understanding. Whereas these original goals are still
open, connections from cluster algebras to various other  areas in
Mathematics have been discovered since. We explain how cluster algebras
arise from triangulations of bordered surfaces.