  Let $X, Y$ be complete metric spaces and $E, F$ be Banach spaces. A bijective
linear operator from a space of $E$-valued functions on $X$ to a space of
$F$-valued functions on $Y$ is said to be biseparating if $f$ and $g$ are
disjoint if and only if $Tf$ and $Tg$ are disjoint. We introduce the class of
generalized Lipschitz spaces, which includes as special cases the classes of
Lipschitz, little Lipschitz and uniformly continuous functions. Linear
biseparating maps between generalized Lipschitz spaces are characterized as
weighted composition operators, i.e., of the form $Tf(y) = S_y(f(h^{-1}(y))$
for a family of vector space isomorphisms $S_y: E \to F$ and a homeomorphism $h
: X\to Y$. We also investigate the continuity of $T$ and related questions.
Here the functions involved (as well as the metric spaces $X$ and $Y$) may be
unbounded. Also, the arguments do not require the use of compactification of
the spaces $X$ and $Y$.
