The classical Riemann zeta function encodes deep information about the distribution of prime numbers, and the Riemann Hypothesis concerns the zeros of this function. The values of the zeta function at even integers are understood (e.g. zeta(2)=(pi^2)/6), but the values at odd integers are mysterious. This talk concerns a several-variable generalization of the zeta function, called a multizeta function, and considers its values from a combinatorial point of view. (The geometric and arithmetic aspects of the theory will be discussed in two seminar talks in the same week.)