  We develop the Riemann-Hilbert problem approach to inverse scattering for the
two-dimensional Schrodinger equation at fixed energy. We obtain global or
generic versions of the key results of this approach for the case of positive
energy and compactly supported potentials. In particular, we do not assume that
the potential is small or that Faddeev scattering solutions do not have
singularities (i.e. we allow the Faddeev exceptional points to exist).
Applications of these results to the Novikov-Veselov equation are also
considered.
