  The paper contains general results on the uniqueness of a DG enhancement for
triangulated categories. As a consequence we obtain such uniqueness for the
unbounded categories of quasi-coherent sheaves, for the triangulated categories
of perfect complexes, and for the bounded derived categories of coherent
sheaves on quasi-projective schemes. If a scheme is projective then we also
prove a strong uniqueness for the triangulated category of perfect complexes
and for the bounded derived categories of coherent sheaves. These results
directly imply that fully faithful functors from the bounded derived categories
of coherent sheaves and the triangulated categories of perfect complexes on
projective schemes can be represented by objects on the product.
