  Classical theorem of Luzin states that a measurable function of one real
variable is "almost" continuous. For measurable functions of several variables
the analogous statement (continuity on the product of sets having almost full
measure) does not hold in general. Searching for a right analogue of Luzin
theorem leads to a notion of virtually continuous functions of several
variables. This probably new notion implicitly appears in the statements of
embedding theorems and trace theorems for Sobolev spaces. In fact it reveals
the nature of such theorems as statements about virtual continuity. Our results
imply that under conditions of Sobolev theorems there is a well-defined
integration of a function over wide class of singular measures, including the
measures concentrated on submanifolds. The notion of virtual continuity is used
also for the classification of measurable functions of several variables and in
some questions on dynamical systems, theory of polymorphisms and bistochastic
measures. In this paper we recall necessary definitions and properties of
admissible metrics, give several definitions of virtual continuity and discuss
some applications. Revised version (without the proofs) is published in
\cite{VZPFA}.
