The theory of multiple zeta values consists in the algebraic and geometric
study of the values at positive integers of
many-variabled -functions.
These numbers satisfy a double family of fundamental algebraic relations
called "double shuffle relations". In this
lecture, we will pose some of the main questions facing the theory at present,
and give some of the major results. Then we will cover the astonishing
connections between the double shuffle algebra and many other parts of
mathematics: moduli spaces of curves and mixed Tate motives, modular forms and
the Eichler-Shimura correspondence, and Grothendieck-Teichmüller theory.
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