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… Expand