  We construct a class of discontinuous superprocesses with dependent spatial
motion and general branching mechanism. The process arises as the weak limit of
critical interacting-branching particle systems where the spatial motions of
the particles are not independent. The main work is to solve the martingale
problem. When we turn to the uniqueness of the process, we generalize the
localization method introduced by [D.W. Stroock, Diffusion processes associated
with Levy generators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32(1975)
209--244] to the measure-valued context. As for existence, we use particle
system approximation and a perturbation method. This work generalizes the model
introduced in [D.A. Dawson, Z. Li, H. Wang, Superprocesses with dependent
spatial motion and general branching densities, Electron. J. Probab. 6(2001),
no.25, 33 pp. (electronic)] where quadratic branching mechanism was considered.
We also investigate some properties of the process.
