  In this paper we study relations between almost specification property,
asymptotic average shadowing property and average shadowing property for
dynamical systems on compact metric spaces. We show implications between these
properties and relate them to other important notions such as shadowing,
transitivity, invariant measures, etc. We provide examples that compactness is
a necessary condition for these implications to hold. As a consequence of our
methodology we also obtain a proof that limit shadowing in chain transitive
systems implies shadowing.
