Multiple polylogarithms Li_{k_1,\ldots,k_d}(x_1,\ldots,x_d) are a class of multi-variable special functions appearing in connection with K-theory, hyperbolic geometry, values of zeta functions/L-functions/Mahler measures, mixed Tate motives, and in high-energy physics.

One of the main challenges in the study of multiple polylogarithms resolves around understanding how on many variables a multiple polylogarithm function (or `interesting' combinations thereof) actually depend (``the depth''), as for example Li_{1,1} can already be expressed via Li_2. Goncharov gave a conjectural criterion (``the Depth Conjecture'') for determining this, using the motivic coproduct, as part of his programme to investigate Zagier's Polylogarithm Conjecture on values of the Dedekind zeta function \zeta_F(m).

I will give an overview multiple polylogarithms, Goncharov's Depth Conjecture, and its implications. I will discuss what is currently known, and what we are still trying to investigate.