The author introduces the notions of partly smooth functions and partly smooth sets together with illustrative examples including the pointwise maximum of some smooth functions and the maximum eigenvalue of a parametrized symmetric matrix. He gives some properties of such functions and sets in terms of subderivatives and subgradients due to R. T. Rockafellar and R. J.-B. Wets [Variational analysis (Grundlehren der Mathematischen Wissenschaften 317, Springer, Berlin) (1998; Zbl 0888.49001)]. He shows invariant properties of the class of partly smooth functions by proving a variety of calculus rules. He also shows under a suitable regularity condition that critical points of partly smooth functions are stable on active manifolds, and develops the connection of partly smoothness with the \({\mathcal U}\)-Lagragian theory for convex functions.