  In this paper we shall develop a theory of (extended) double shuffle
relations of Euler sums which generalizes that of multiple zeta values (see
Ihara, Kaneko and Zagier, \emph{Derivation and double shuffle relations for
multiple zeta values}. Compos. Math. \textbf{142} (2)(2006), 307--338). After
setting up the general framework we provide some numerical evidence for our two
main conjectures. At the end we shall prove the following long standing
conjecture: for every positive integer n
$$\zeta(\{3\}^n)=8^n\zeta(\{\ol2,1\}^n).$$ The main idea is to use the double
shuffle relations and the distribution relation. This particular distribution
relation doesn't follow from the double shuffle relation in general. But we
believe it does follow from the extended double shuffle relations.
