These are notes from introductory survey lectures given
at the Institute for Studies in Theoretical Physics and Mathematics
(IPM), Teheran, in 2008 and 2010. We present the definition and
the fundamental properties of Fomin-Zelevinsky’s cluster algebras.
Then, we introduce quiver representations and show how they can
be used to construct cluster variables, which are the canonical generators
of cluster algebras. From quiver representations, we proceed
to the cluster category, which yields a complete categorification of
the cluster algebra and its combinatorial underpinnings.