  Semi-Riemannian manifolds that satisfy (homogeneous) linear differential
conditions of arbitrary order on the curvature are analyzed. They include, in
particular, the spaces with (higher-order) recurrent curvature, (higher-order)
symmetric spaces, as well as entire new families of semi-Riemannian manifolds
rarely, or never, considered before in the literature --such as the spaces
whose derivative of the Riemann tensor field is recurrent, among many others.
Definite proof that all types of such spaces do exist is provided by exhibiting
explicit examples of all possibilities in all signatures, except in the
Riemannian case with a positive definite metric. Several techniques of
independent interest are collected and presented. Of special relevance is the
case of Lorentzian manifolds, due to its connection to the physics of the
gravitational field. This connection is discussed with particular emphasis on
Gauss-Bonnet gravity and in relation with Penrose limits. Many new lines of
research open up and a handful of conjectures, based on the results found
hitherto, is put forward.
