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 omega-limit sets of the closed-loop system with the speed-gradient (or "Jurdjevic-Quinn") controllers for "affinized" controlled system. Two illustrating examples are given.