  In this paper we define an explicit E_{infinity}-structure, i.e. a coherently
homotopy associative and commutative product on chain complexes defining
(integral and mod-l) motivic cohomology as well as mod -l \'etale cohomology.
We also discuss several applications. In addition, our constructions show that
the source of the E_{\infinity}-structure on the motivic complexes provided
with the pairing defined by Suslin and Voevodsky is not chain-theoretic as is
the case for the singular co-chain complexes for topological spaces.
