Classical polylogarithms have been studied extensively since pioneering work of Euler and Abel. It is known that they satisfy lots of functional equations, but in weight >4 these equations are not known yet. Even in the weight 4 they were first found using heavy computer-assisted computations. 

The main goal of the talk is to explain the depth conjecture for polylogarithms and its relation to functional equations and the Zagier conjecture about special values of zeta functions. It is proved in weight 4, and the proof is based on some new ideas from the theory of cluster algebras and Poisson geometry. 

The talk is based on joint work with A. Goncharov.