Abstract: Zeta values and multiple zeta values are positive real numbers investigated by Euler in the late eighteenth century. Surprisingly, these numbers are ubiquitous in many mathematical and physical theories such as number theory, mixed Tate motives, and quantum field theory.

By a well-known analogy between the arithmetic of number fields and that of function fields of positive characteristic, Carlitz and Thakur introduced a theory of positive characteristic zeta values and multiple zeta values associated to the projective line. During the last twenty years, many people have developed this theory in several directions, have proved important theorems and have formulated several conjectures. In this talk, we will explain recent developments towards these conjectures.