  We define the twisted tensor product of two enriched categories, which
generalizes various sorts of `products' of algebraic structures, including the
bicrossed product of groups, the twisted tensor product of (co)algebras and the
double cross product of bialgebras. The key ingredient in the definition is the
notion of simple twisting systems between two enriched categories. To give
examples of simple twisted tensor products we introduce matched pairs of
enriched categories. Several other examples related to ordinary categories,
posets and groupoids are also discussed.
