We prove the periodicities of the restricted T and Y-systems associated with
the quantum affine algebra of type B_r at any level. We also prove the
dilogarithm identities for the Y-systems of type B_r at any level. Our proof is
based on the tropical Y-systems and the categorification of the cluster algebra
associated with any skew-symmetric matrix by Plamondon. Using this new method,
we also give an alternative and simplified proof of the periodicities of the T
and Y-systems associated with pairs of simply laced Dynkin diagrams.

This is an introduction to some aspects of Fomin-Zelevinsky's cluster
algebras and their links with the representation theory of quivers and with
Calabi-Yau triangulated categories. It is based on lectures given by the author
at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to
by now classical material, we present the outline of a proof of the periodicity
conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and
recent results on the interpretation of mutations as derived equivalences.