The inverse scattering method captures long-term behavior of completely integrable, dispersive partial differential equations in two dimensions, at the cost of restrictive conditions on the initial data. These restrictions involve the spectrum of the associated linear operator that defines the scattering transform, and rule out such phenomena as soliton solutions or solutions which blow up in finite time. In this talk I'll discuss recent progress, open problems and possible approaches, using the focussing Davey-Stewartson equations and the Novikov-Veselov equation as examples. A key role in the analysis is played by the renormalized determinant of the linear operator that defines the scattering transform.