This chapter presents the notions of higher stability and absolute stability for semigroup actions. It presents the concepts of higher prolongation and higher prolongational limit set. They are respectively used to formalize the concepts of absolute stability and Auslander generalized recurrence. The higher prolongation extends the forward orbit and determines the higher stability. A set that is all-order stable is called absolute stable. The notion of Auslander generalized recurrent point extends the concept of nonwandering point and is a particular type of chain recurrence. This chapter also gives some applications to semigroup of operators in Banach function spaces, multi-time dynamical systems, and control affine systems.