  We introduce and study spaces of multivariate functions of bounded variation
generalizing the classical Jordan and Wiener spaces. Multivariate
generalizations of the Jordan space were given by several prominent researchers
but each of them preserved only some special properties of the space used
further in few selected applications. Unlike this the multivariate
generalization of the Jordan space presented in this paper preserves all known
and reveals some previously unknown properties of the space. These, in turn,
are special cases of the basic properties of the introduced spaces proved in
the paper. They, in particular, include results on discontinuity sets and
pointwise differentiability of the bounded variation functions and their Luzin
type and $C^\infty$ approximation. Moreover, the second part of the paper
presents results on Banach structure of function spaces of bounded variation,
namely, atomic decomposition and constructive characterization of their predual
spaces and then constructive characterization of preduals of the last ones and
following from here the so-called two-stars theorems relating second duals of
separable subspaces of "vanishing variation" to the (nonseparable) spaces of
bounded variation.
