Multiple zeta values (MZVs) and their generalizations, i.e., multiple polylogarithms (MPLs) have recently attracted much attention, both in pure mathematics and theoretical physics. A systematic study only started in the early 1990s, although the prehistory can be traced back to Euler in the 18th century. Research on MZVs and MPLs comprises several mathematical areas, including algebra (Hopf and Lie algebras), combinatorics (double shuffle relations), algebraic geometry (Grothendieck's motives), Lie group theory (Kashiwara-Vergne conjecture), and, of course, number theory. Moreover, they have deep connections with high-energy physics (Feynman diagrams in perturbative quantum field theory).

This series of lectures consists of 3 mini-courses. It aims at introducing in pedagogical lectures, accessible to advanced master and early PhD students, the fascinating theory of MZVs and MPLs. The lecturers are going to present the basic structures and its generalizations as well as its ramifications into other areas of modern mathematics.