In this paper, we investigate set optimization problems with three types of set order relations. Various kinds of well-posedness for these problems and their relationship are concerned. Then, sufficient conditions for set optimization problems to be well-posed are established. Moreover, Kuratowski measure of noncompactness is applied to survey characterizations of well-posedness for set optimization problems. Furthermore, approximating solution maps and their stability are researched to propose the link between stability of the approximating problem and well-posedness of the set optimization problem.