Biomagnetic imaging is the estimation and display of current density distributions calculated from measurements of the magnetic fields they generate. The magnetic fields measurements alone, no matter how dense and extensive, are not sufficient for uniquely reconstructing the current densities. In this paper we concentrate on quantifying the limitations and on introducing constraints into the reconstruction techniques. We apply the technique of characteristic bodies as transplanted from the microwave and radar theory. This technique allows for simple calculations when introducing a particular class of constraints, i.e., limiting sources to surfaces. In particular, we look at surface current and surface monopole distributions that generate magnetic fields corresponding to the measured values. We also show how magnetic fields known on arbitrary closed surfaces may be projected to other surfaces and then the characteristic body technique used to calculate source distributions. In addition we consider differential and support constraints.