abstract: Periods are complex numbers that can be represented as integrals of algebraic differential forms along topological cycles on algebraic varieties. According to conjectures of Grothendieck and Kontsevich-Zagier, all polynomial relations between these numbers should have "geometric origin". This would in particular give rise to a Galois theory of periods generalising the usual Galois action on algebraic numbers. I will explain how our progress in the understanding of motives over the last twenty years has allowed for two equivalent, albeit very different unconditional constructions of the motivic Galois group. Among their striking applications, I will sketch the main ideas in Ayoub's proof of a relative version of the Kontsevich-Zagier conjecture, where periods are roughly speaking replaced by series of periods. Hodge theory enters the stage through the theorem of the fixed part.