Abstract: We discuss conformal, spherical CR-geometries and connected with them symmetric rank one negatively curved geometries on manifolds, as well as dynamics of discrete action of their fundamental groups in such spaces (dynamics of their orbits as well as the ergodicity of their action at the spheres at infinity, related dynamical systems, and dynamics of equivariant mappings with bounded distortion). This is closely related to algebra of varieties of discrete group representations (Teichmüller spaces of manifolds), topology of manifolds and geometric analysis in corresponding spaces. A unifying theme is the discrete holonomy groups of the corresponding geometric structures. We should avoid technicalities as much as Jeff Weeks demonstrated in his famous “bed time reading” book “The shape of space”.