  We study the de Rham fundamental group of the configuration space $E^{(n)}$
of $n+1$ marked points on an elliptic curve $E$, and define multiple elliptic
polylogarithms. These are multivalued functions on $E^{(n)}$ with unipotent
monodromy, and are constructed by a general averaging procedure.
  We show that all iterated integrals on $E^{(n)}$, and in particular the
periods of the unipotent fundamental group of the punctured curve $E \backslash
\{0\}$, can be expressed in terms of these functions.
