The paper deals with the proof method of verification by augmented finitary abstraction (VAA), which presents an effective approach to the verification of the temporal properties of (potentially infinite-state) reactive systems. The method consists of a two-step process by which, in a first step, the system and its temporal specification are combined an then abstracted into a finite-state Biichi automaton. The second step uses model checking to establish emptiness of the abstracted automaton. The VAA method can be considered as a viable alternative to verification by temporal deduction which, up to now, has been the main method shown to be complete for the verification of infinite-state systems. The paper presents a general recipe for the abstraction of Buchi automata which is shown to be sound, where soundness means that emptiness of the abstract automaton implies emptiness of the concrete (infinite-state) automaton. To make the method applicable for the verification of liveness properties, pure abstraction is sometimes no longer adequate. We show that by augmenting the system by an appropriate (and standardly constructible) progress monitor, we obtain an augmented system, whose computations are essentially the same as the original system, and which may now be abstracted while preserving the desired liveness properties. We then proceed to show that the VAA method is sound and complete for proving all properties expressible by temporal logic (including both safety and liveness). Completeness establishes that whenever an infinite-state Biichi automaton has no computations, there exists a finitary abstraction which abstracts the automaton, augmented by an appropriate progress monitor, into a finite-state Biichi automaton with no computations.