Virtually all textbook treatments of jump-diffusion SDEs assume that the driving processes are a Brownian motion and an independent homogeneous Poisson random measure. In many applications, for example modelling of credit-risky securities, it seems that solution-dependence of the compensator of the random measure should be allowed. The reason for not including this goes back to the 1972 book of Gihman and Skorohod, where it is shown how a problem with state-dependent compensator can be 'reduced' to an equivalent one with homogeneous random measure. There may however be good reasons for not doing this transformation: for example the homogeneous random measure may have infinite activity even if the jump rate in the original model is a.s. finite. These questions are discussed and some general results about existence and uniqueness with state-dependent jump measure are given.