Exterior differential ideals are discussed, and sets of invariant generators presented, for Riemannian, conformal and projective geometries, and for specialisations such as Ricci-flat, self-dual and Einstein-Maxwell theories. The Cartan characteristic integers are explicitly calculated, and involutory basis forms found, for each of these (specialised to four dimensions), exposing their algebraic structure and showing how they generate well-posed sets of partial differential equations.