  We establish a new class of relations among the multiple zeta values
\zeta(k_1,k_2,...,k_n), which we call the cyclic sum identities. These
identities have an elementary proof, and imply the "sum theorem" for multiple
zeta values. They also have a succinct statement in terms of "cyclic
derivations" as introduced by Rota, Sagan and Stein. In addition, we discuss
the expression of other relations of multiple zeta values via the shuffle and
"harmonic" products on the underlying vector space H of the noncommutative
polynomial ring Q<x,y>, and also using an action on Q<x,y> of the Hopf algebra
of quasi-symmetric functions.
