  The paper sketches a recent progress and formulates several open problems in
studying equivariant quasiconformal and quasisymmetric homeomorphisms in
negatively curved spaces as well as geometry and topology of noncompact
geometrically finite negatively curved manifolds and their boundaries at
infinity having Carnot--Carath\'eodory structures. Especially, the most
interesting are complex hyperbolic manifolds with Cauchy--Riemannian structure
at infinity, which occupy a distinguished niche and whose properties make them
surprisingly different from real hyperbolic ones.
