Valeurs z\^eta multiples

This survey article is the written version of two talks given at the Journ'ees X-UPS 2019 “P'eriodes et transcendance” at 'Ecole polytechnique. We give a gentle introduction to the study of multiple zeta values, from Euler’s solution to the Basel problem to modern Galois theoretic tools. We emphasize the role of the theory of motives which conjecturally controls the structure of the algebra of multiple zeta values.