By the great importance of quasiequilibrium models in practical applications, stability and well-posedness conditions for them have motivated and inspired studies of many mathematicians. In most works devoted to this topic, topological properties of constraint sets are imposed directly on the fixed points of the set-valued maps, and hence, it is difficult to check them when sufficient conditions for stability and well-posedness are in use. To avoid this drawback, in this chapter, we first study separately many important properties of solution maps of fixed-point problems of set-valued maps and then employ these results to the aforementioned stability and well-posedness conditions directly in terms of the problem data. Furthermore, we also give an overview of sufficient conditions for Hölder and Lipschitz properties of solution maps of quasiequilibrium problems via assumptions related to monotonicity and convexity. Furthermore, under relaxed conditions, the calmness property of such maps is also derived in this chapter. Note that for solution maps of quasiequilibrium problems, the difference between stability and well-posedness in the sense of semicontinuity and continuity and in the sense of Hölder and Lipschitz properties is so big that many approaches and techniques for dealing with equilibrium problems cannot be employed. As a result, several established results for equilibrium problems have not yet been extended to quasiequilibrium problems, and hence, deriving Hölder and Lispchitz conditions for the solution maps of such models is still an open problem.