  We study the dilogarithm identities from algebraic, analytic, asymptotic,
$K$-theoretic, combinatorial and representation-theoretic points of view. We
prove that a lot of dilogarithm identities (hypothetically all !) can be
obtained by using the five-term relation only. Among those the Coxeter, Lewin,
Loxton and Browkin ones are contained. Accessibility of Lewin's one variable
and Ray's multivariable (here for $n\le 2$ only) functional equations is given.
For odd levels the $\hat{sl_2}$ case of Kuniba-Nakanishi's dilogarithm
conjecture is proven and additional results about remainder term are obtained.
The connections between dilogarithm identities and
Rogers-Ramanujan-Andrews-Gordon type partition identities via their asymptotic
behavior are discussed. Some new results about the string functions for level
$k$ vacuum representation of the affine Lie algebra $\hat{sl_n}$ are obtained.
Connection between dilogarithm identities and algebraic $K$-theory (torsion in
$K_3({\bf R})$) is discussed. Relations between crystal basis, branching
functions $b_{\lambda}^{k\Lambda_0}(q)$ and Kostka-Foulkes polynomials
(Lusztig's $q$-analog of weight multiplicity) are considered. The Melzer and
Milne conjectures are proven. In some special cases we are proving that the
branching functions $b_{\lambda}^{k\Lambda_0}(q)$ are equal to an appropriate
limit of Kostka polynomials (the so-called Thermodynamic Bethe Ansatz limit).
Connection between "finite-dimensional part of crystal base" and
Robinson-Schensted-Knuth correspondence is considered.
