Many geometries may be viewed as curved versions of F. Klein's homogeneous models and many of them allow a unified and effective treatment. This had been one of the outputs of the famous E. Cartan's equivalence method about hundred years ago. The later development led to the concepts of principal bundles and connections, and G-structures but many of the striking features of the original concept of `espace generalisé' (now called Cartan connection) got nearly forgotten.
The talk will review some basic examples like conformal Riemannian, projective, and CR geometries in the realm of a recent general calculus. Only elementary Lie theory and differential geometry is used in order to obtain a powerful machinery for geometrical and analytical problems. The applications of these methods in the recent paper `A. Cap, J. Slovak., V. Soucek, Bernstein-Gelfand-Gelfand Sequences, Annals of Math. 154 (2001), 97-113' will be touched in the end of the lecture.)