  For complete metric spaces $X$ and $Y$, a description of linear biseparating
maps between spaces of vector-valued Lipschitz functions defined on $X$ and $Y$
is provided. In particular it is proved that
  $X$ and $Y$ are bi-Lipschitz homeomorphic, and the automatic continuity of
such maps is derived in some cases. Besides, these results are used to
characterize the separating bijections between scalar-valued Lipschitz function
spaces when $Y$ is compact.
