The paper concerns a systematic study of full stability in general optimization models including its conventional Lipschitzian version as well as the new Hölderian one. We derive various characterizations of both Lipschitzian and Hölderian full stability in nonsmooth optimization, which are new in finite-dimensional and infinite-dimensional frameworks. The characterizations obtained are given in terms of second-order growth conditions and also via second-order generalized differential constructions of variational analysis. We develop effective applications of our general characterizations of full stability to conventional models of nonlinear programming, to optimization problems with polyhedric constraints in infinite dimensions, and to optimal control problems governed by semilinear elliptic PDEs.