Multiple Zeta Values and Their Extended Family

A multiple zeta value (MZV) is a real number specified by a string of positive integers.
The sum of the string is the weight, and its length is the depth.  MZVs of depth one are
just values of the Riemann zeta function at positive integers; MZVs of depth two were
already studied by Euler.  MZVs of arbitrary depth became a hot topic in the early 1990s
with their simultaneous appearance in perturbative quantum field theory and knot
theory.  Since then the field has continued to develop rapidly.  I will describe various
results about MZVs and their relatives, such as alternating ("colored") MZVs and multiple
t-values.