  We consider relationships between families of K3 surfaces, in the context of
string theory. An important ingredient of string theory also of interest in
algebraic geometry is T-duality. Donagi and Pantev have extended the original
duality on genus one fibred K3 surfaces with a section to the case of any genus
one fibration, via a Fourier-Mukai transform. We investigate possibilities of
extending this result to the more general case of non fibred K3s, in the
context of Caldararu's conjecture.
  We also provide an explicit construction of a quasi-universal sheaf
associated to the classical Mukai correspondence.
