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The paper is a general overview of an approach to the semantics of logic programs whose aim is nding notions of models which really capture the operational semantics, and are therefore useful for de ning program equivalences and for semantics-based program analysis. The approach leads to the introduction of extended interpretations which are more expressive than Herbrand interpretations. The semantics in terms of extended interpretations can be obtained as a result of both an operational (top-down) and a xpoint (bottom-up) construction. It can also be characterized from the model-theoretic viewpoint, by de ning a set of extended models which contains standard Herbrand models. We discuss the original construction modeling computed answer substitutions, its compositional version and various semantics modeling more concrete observables. We then show h o w the approach can be applied to several extensions of positive logic programs. We nally consider some applications, mainly in the area of semantics-based program transformation and analysis.