Instigated by work of Borel and Bloch, Zagier formulated his Polylogarithm Conjecture in the late eighties and proved it for weight 2. After a flurry of activity and advances at the time, notably by Goncharov who not only provided powerful new tools for a proof in weight 3 but also set out a vast program with a plethora of conjectural statements for attacking it, progress seemed to be stalled for a number of years. More recently, a solution to one of Goncharov's central conjectures in weight 4 has been given. Moreover, by adopting a new point of view, work by Goncharov and Rudenko gave a proof of the original conjecture in weight 4. In this impressionist talk I intend to give a rough idea of the developments from the early days on, avoiding most of the technical bits, and also hint at a number of recent results for higher weight (joint with S.Charlton and D.Radchenko).