Differential Equations Driven by Rough Paths : École dÉté de Probabilités de Saint Flour XXXIV 2004 Each year young mathematicians congregate in Saint Flour France and listen to extended lecture courses on new topics in Probability Theory. The goal of these notes representing a course given by Terry Lyons in 2004 is to provide a straightforward and self supporting but minimalist account of the key results forming the foundation of the theory of rough paths. The proofs are similar to those in the existing literature but have been refined with the benefit of hindsight. The theory of rough paths aims to create the appropriate mathematical framework for expressing the relationships between evolving systems by extending classical calculus to the natural models for noisy evolving systems which are often far from differentiable. Differential Equations Driven by Moderately Irregular Signals The Signature of a Path Rough Paths Integration Along Rough Paths Differential Equations Driven by Rough Paths.