  In this paper we present some of the recent progresses in multiple zeta
values (MZVs). We review the double shuffle relations for convergent MZVs and
summarize generalizations of the sum formula and the decomposition formula of
Euler for MZVs. We then discuss how to apply methods borrowed from
renormalization in quantum field theory and from pseudodifferential calculus to
partially extend the double shuffle relations to divergent MZVs.
