We propose definitions of well-posedness under perturbation in terms of suitable asymptotically solving sequences, not
embedding a given problem into a parameterized family. When considering in detail quasi-equilibrium and quasi-optimization problems as well as optimization problems with equilibrium constraints, we establish sufficient conditions for
both well-posedness and unique well-posedness. For cases with noncompact underlying sets of constraints, measures of
noncompactness are employed. We provide numerous examples to ensure the essentialness of the assumptions imposed in the obtained results.