  Multiple zeta values (MZVs, also called Euler sums or multiple harmonic
series) are nested generalizations of the classical Riemann zeta function
evaluated at integer values. The fact that an integral representation of MZVs
obeys a shuffle product rule allows the possibility of a combinatorial approach
to them. Using this approach we prove a longstanding conjecture of Don Zagier
about MZVs with certain repeated arguments. We also prove a similar cyclic sum
identity. Finally, we present extensive computational evidence supporting an
infinite family of conjectured MZV identities that simultaneously generalize
the Zagier identity.
