Abstract: The appearance of multiple zeta values in Theoretical Physics as well as in computations of Drinfel'd and Deligne stimulated great interest. One strategy leading to a better understanding of these numbers has been to study the polylogarithm functions (which go back to Euler). Since these functions admit expressions as both sums and iterated integrals, they give rise to two distinct Hopf algebra structures. It is believed that all polynomial relations satisfied by such numbers are known, and arise from these structures. In this talk, we outline this theory, and introduce the use of complex iterated integrals to define a Hopf algebra structure on polylogarithm functions at non-integer values.