Rectifiable sets are generalizations of smooth manifolds and form the central objects of geometric measure theory. They are widely studied in several areas of application for their geometric properties such as an admittance of a notion of tangent bundles and their analytic properties which include their amenability to Lipschitz maps and availability of a Gauss-Green theorem, area and co-area formulas.

In this talk, we will introduce this interesting class of sets essentially talking about their geometry and some analysis on them which inform their ubiquity in several fields. We also discuss well-known characterizations of these sets and, if time permits, their higher order analogues.