Biseparating maps between Lipschitz function spaces

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.