Trying to understand cosmological anomalies (dark energy, dark matter), many physicists consider various generalizations of the General Relativity Theory. E.g.: theories derived from a Lagrangian depending not only upon the curvature tensor, but also upon its higher covariant derivatives. Personally, I do not believe in the physical relevance of such theories. But, when analizing their mathematical structure, one discovers a beautiful "Terra Nova" of geometric constructions, which sheds also new light on the classical notion of curvature.