There are many parallels between the theories of the integers and the polynomial ring in one variable over a finite field. In the 1930's Carlitz constructed function field valued analogues of the Riemann zeta function, and in 1980's Goss greatly generalized Carlitz's zeta function to L-functions associated to Drinfeld modules. More recently Thakur has defined multiple zeta values in positive characteristic in line with classical multiple zeta values of Euler. In the classical case there are many connections between special values of these analytic functions and arithmetic invariants of their underlying algebraic structures, and it is a natural question to ask to what extent these expectations hold true in positive characteristic. In spite of tantalizing examples, this remained a mystery for many years, until Taelman proved a class number formula for special values in 2012.

In this talk we will survey the classical theory of Riemann zeta values and multiple zeta values, as well as the history of Goss L-series. This will lead to discussion of the advances of Taelman, as well as further directions defined by Thakur on multiple zeta values and Pellarin on deformations of Goss L-series. We will also present results on Eulerian multiple zeta values and log-algebraic identities for L-series attached to Drinfeld modules (joint with C.-Y. Chang, A. El-Guindy, and J. Yu).