The mathematical foundations of invariant signatures for object recognition and symmetry detection are based on the Cartan theory of moving frames and its more recent extensions developed with a series of students and collaborators. The moving frame calculus leads to mathematically rigorous differential invariant signatures for curves, surfaces, and moving objects. The theory is readily adapted to the design of noise-resistant alternatives based on joint (or semi-)differential invariants and purely algebraic joint invariants. Such signatures can be effectively used in the detection of exact and approximate symmetries, as well as recognition and reconstruction of partially occluded objects. Moving frames can also be employed to design symmetry-preserving numerical approximations to the required differential and joint differential invariants.