  An additive map $T$ acting between spaces of vector-valued functions is said
to be biseparating if $T$ is a bijection so that $f$ and $g$ are disjoint if
and only if $Tf$ and $Tg$ are disjoint. Note that an additive bijection retains
$\mathbb{Q}$-linearity. For a general nonlinear map $T$, the definition of
biseparating given above turns out to be too weak to determine the structure of
$T$. In this paper, we propose a revised definition of biseparating maps for
general nonlinear operators acting between spaces of vector-valued functions,
which coincides with the previous definition for additive maps. Under some mild
assumptions on the function spaces involved, it turns out that a map is
biseparating if and only if it is locally determined. We then delve deeply into
some specific function spaces -- spaces of continuous functions, uniformly
continuous functions and Lipschitz functions -- and characterize the
biseparating maps acting on them. As a by-product, certain forms of automatic
continuity are obtained. We also prove some finer properties of biseparating
maps in the cases of uniformly continuous and Lipschitz functions.
