Quasiconformal mappings and asymptotic geometry of manifolds

We will give a commentary on the origins of the theory of quasiconformal mappings, and will talk about multidimensional quasiconformal mappings and relations with the conformal geometry of Riemannian manifolds. We will discuss the global homeomorphism theorem in initial and modern versions. In this connection we will discuss the conformal classification of Riemannian manifolds, the geometric features of the conformal type of a manifold and reduction of the isoperimetric function of a manifold to the normal form by conformal change of the Riemannian metric. Finally we plan to formulate several open problems related to the topic under discussion.