According to J. Hadamard's famous statement, an equation is well-posed if the following are satisfied: i) there exists a solution, ii) the solution is unique, iii) the solution depends continuously on the initial data. In this talk we carry out the three tasks of this program for BSDEs with jumps. More specifically, in the first part of this talk we will provide existence and uniqueness results for BSDEs with jumps driven by martingales that are stochastically discontinuous, hence we can treat BSDEs and BSΔEs in a unified and general framework. Then, we will present stability results for martingale representations. The final part consists of stability results for solutions of BSDEs not only with respect to the initial data, but also with respect to discretized versions of the driving martingale.