  We study U(r) instantons on elliptic surfaces with a section and show that
they are in one-one correspondence with spectral data consisting of a curve in
the dual elliptic surface and a line bundle on that curve. We use relative
Fourier-Mukai transforms to analyse their properties and, in the case of the K3
and abelian surfaces, we show that the moduli space of instantons has a natural
Lagrangian fibration with respect to the canonical complex symplectic
structure.
