Abstract: Multiple polylogarithms are often defined initially as multiply nested series. They reduce to the classical polylogarithm function in the case of only one level of nesting. By specializing the arguments in multiple polylogarithms, we obtain multiple zeta values, a nested series extension of the Riemann zeta function of interest in knot theory and quantum field theory as well as algebra, number theory, and combinatorics. Many interesting relations exist between multiple polylogarithms of various sorts. A good many relations were discovered using sophisticated computational techniques, and some of these have subsequently been proven. In this non-technical survey, I'll discuss some of the broader questions arising in the study of multiple polylogarithms and multiple zeta values, why researchers find these questions interesting, and what methods are being employed in attempting to answer them.