  This is a reasonably self-contained exposition of the fascinating interplay
between cluster algebras and the dilogarithm in the recent two decades. The
dilogarithm has a long and rich history since Euler. The most intriguing
property of the function is that it satisfies various functional relations,
which we call dilogarithm identities (DIs). In the 1990s Gliozzi and Tateo
conjectured a family of DIs based on root systems, which vastly generalize the
classic DIs such as Euler's and Abel's identities. On the other hand, the
cluster algebras are a class of commutative algebras introduced by Fomin and
Zelevinsky around 2000. It turned out that a DI is associated with each period
in a cluster pattern of cluster algebra theory. In particular, they solve the
conjecture on the DIs by Gliozzi and Tateo. We employ these DIs as the
leitmotif and present several proofs, variations, and generalizations of them
with various methods and techniques. The quantum DIs are also treated from the
unified point of view with the classical ones.
