The field of rough analysis provides a powerful mathematical framework for understanding differential equations driven by irregular paths, which frequently arise in the modeling of complex real-world phenomena. Studying such equations requires an understanding of integration against paths, a problem that becomes increasingly challenging as the regularity of the path decreases. This motivates the fundamental notions of a rough path and the rough integral, pioneered by Terry Lyons in the 1990s, which are central for the theory of rough analysis. In this talk, we aim to give a gentle introduction to these concepts, and show how they lead to a robust theory for rough differential equations. We demonstrate consistency with Itô's stochastic calculus for Brownian paths, and illustrate how rough analysis addresses limitations in the theory of stochastic differential equations.