The paper deals with the proof method of verification by finitary abstraction (VFA), which presents a feasible 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 jointly abstracted into a finite-state system and a finite-state specification. The second step uses model checking to establish the validity of the abstracted property over the abstracted system. The VFA method can be considered a viable alternative to verification by temporal deduction which, up to now, has been the main method generally applicable for verification of infinite-state systems. The paper presents a general recipe for the joint abstraction, which is shown to be sound, where soundness means that validity over the abstract system implies validity over the concrete (original) system. 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 refer to the extended method as verification by augmented abstraction (VAA). 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 the property is valid, there exists a finitary abstraction which abstracts the system, augmented by an appropriate progress monitor, into a finite-state system which validated the abstracted property.