  The behaviour of supersymmetric D-branes under deformations of the closed
string background is studied using world-sheet methods. We explain how lines of
marginal stability and obstructions arise from this point of view. We also show
why N=2 B-type branes may be obstructed against (cc) perturbations, but why
such obstructions do not occur for N=4 superconformal branes at c=6, i.e. for
half-supersymmetric D-branes on K3. Our analysis is based on a field theory
approach in superspace, as well as on techniques from perturbed conformal field
theory.
