  The theory of rough paths arose from a desire to establish continuity
properties of ordinary differential equations involving terms of low
regularity. While essentially an analytic theory, its main motivation and
applications are in stochastic analysis, where it has given a new perspective
on It\^o calculus and a meaning to stochastic differential equations driven by
irregular paths outside the setting of semi-martingales. In this survey, we
present some of the main ideas that enter rough path theory. We discuss
complementary notions of solutions for rough differential equations and the
related notion of path signature, and give several applications and
generalisations of the theory.
