  Multiple zeta values have been studied by a wide variety of methods. In this
article we summarize some of the results about them that can be obtained by an
algebraic approach. This involves "coding" the multiple zeta values by
monomials in two noncommuting variables x and y. Multiple zeta values can then
be thought of as defining a map \zeta: H^0 -> R, where H^0 is the graded
rational vector space generated by the "admissible words" of the noncommutative
polynomial algebra Q<x,y>. Now H^0 admits two (commutative) products making
\zeta a homomorphism: the shuffle product and the "harmonic" product. The
latter makes H^0 a subalgebra of the algebra QSym of quasi-symmetric functions.
We also discuss some results about multiple zeta values that can be stated in
terms of derivations and cyclic derivations of Q<x,y>, and define an action of
the Hopf algebra QSym on Q<x,y> that appears useful. Finally, we apply the
algebraic approach to finite partial sums of multiple zeta value series.
