The problem of the global and local stabilization of invariant sets for general nonlinear controlled systems is considered. New state feedback stabilizing controllers and sufficient conditions of asymptotic stability of a goal set with the specified region of attraction are proposed. The proofs of the obtained results are based on the detailed analysis of the ω-limit sets of the closed-loop system with the speed-gradient (or `Jurdjevic-Quinn') controllers for `affinized' controlled system. Two illustrating examples are given.