  We study the optimal conditions on a homeomorphism $f:\Omega\subset \R^n\to
\R^n$ to guarantee that the composition $u\circ f$ belongs to the space of
functions of bounded variation for every function $u$ of bounded variation. We
show that a sufficient and necessary condition is the existence of a constant
$K$ such that $|Df|(f^{-1}(A))\leq K\Ln(A)$ for all Borel sets $A$. We also
characterize homeomorphisms which maps sets of finite perimeter to sets of
finite perimeter. Towards these results we study when $f^{-1}$ maps sets of
measure zero onto sets of measure zero (i.e. $f$ satisfies the Lusin $(N^{-1})$
condition).
