Conformal and CR geometries are among the class of "parabolic geometries" which posses a canonical Cartan connection characterizing the geometry. Replacing the Levi-Civita connection with the Cartan connection we develop submanifold theory in parallel with the classical Riemannian case. This allows us to apply tools developed for conformal and CR invariant theory to develop a theory of submanifold invariants and invariant operators, relevant to the study of conformally or CR invariant boundary value problems and other problems in geometric analysis involving submanifolds. The technical details of the theory are substantial (especially in the CR case). I will try to emphasize some of the concrete geometric ideas behind the approach, giving insight into the original work of Elie Cartan.