The global conformality of a mapping provide a strong rigidity even on the plane.

In higher dimensions this property holds, in view of the classical Liouville theorem,

only for restrictions of the M\"obius transformations. Thus quasiregular mappings are of greater interest in higher dimensions, since they provide ``the right extension of the geometric parts of the theory of analytic functions in the plane to $\mathbb R^n$'' (Seppo Rickman).

The theory of multidimensional quasiconformal mappings (quasiregular homeomorphisms) employs three main approaches analytic, geometric (moduli) and metric ones.

In the talk, we study the main relationships between various classes of mappings whose definitions rely on metric approaches and techniques: finitely bi-Lipschitz mappings, quasisymmetric mappings, quasim\"obius and quasiconformal mappings, mappings of finite metric and area distortion. The latter is the cental object in our presentation.

Although no analytic restrictions are assumed, some nice and important regularity properties like the Lusin $(N)$-condition, absolute continuity, etc. are derived.

We also involve classes of mappings which are called the ring, lower and hyper $Q$-homeomorphisms and are defined purely geometrically. The interplay between the above classes of mappings allows us to investigate the boundary correspondence problems related to the weakly flat and strongly accessible boundaries on Riemannian manifolds. Several illustrated examples are also presented.

The talk is based on joint works with Elena Afanas'eva (Institute of Applied Mathematics and Mechanics of the NAS of Ukraine).