Progr`es recents sur la conjecture de Zagier et le programme de Goncharov [dapr`es Goncharov, Rudenko, Gangl, ...]

This survey article is the written version of a talk given at the Bourbaki seminar in April 2021. We give an introduction to Zagiers conjecture on special values of Dedekind zeta functions, and its relation to $K$-theory of fields and the theory of motives. We survey recent progress on the conjecture and in particular the proof of the $n=4$ case of the conjecture by Goncharov and Rudenko.