After recalling the arithmetic interest of multiple polylogarithms, I will discuss certain algebraic geometry problems motivated by the study of their elliptic analogues. In particular, I will give some ideas on how the universal vector extension of an elliptic curve allows us to develop an algebraic de Rham theory of multiple elliptic polylogarithms and the KZB equation. This is joint work in progress with Nils Matthes (Copenhagen).