Abstract: Given a smooth path X, the signature of a path is the infinite sequence of the iterated integral of X with itself. If X is a stochastic process with an integration structure (e.g. semimartingale), the corresponding signature is the key-object to understand the "lack of continuity" between the solution of a SDE driven by X and the process itself. In this talk, we will review the main properties of the signature and introduce the new notion of "non-commutative signature", which we tailored to study a new class of rough/stochastic differential equations arising in the context of non-commutative probability. Joint paper with Nicolas Gilliers (Université de Toulouse) and Yannick Vargas (Postdam Universität)