Introduction. Since the originalworks [39, 40] by AbrahamRobinson,many different presentations to the methods of nonstandard analysis have been proposed over the last forty years. The task of combining in a satisfactory manner rigorous theoretical foundations with an easily accessible exposition soon revealed very difficult to be accomplished. The first pioneering work in this direction was W.A.J. Luxemburg's lecture notes [36]. Based on a direct use of the ultrapower construction, those notes were very popular in the “nonstandard” community in the sixties. Also Robinson himself gave a contribution to the sake of simplification, by reformulating his initial typetheoretic approach in a more familiar set-theoretic framework. Precisely, in his joint work with E. Zakon [42], he introduced the superstructure approach, by now the most used foundational framework.

To the authors’ knowledge, the first relevant contribution aimed to make the “hyper-methods” available even at a freshman level, is Keisler's book [33], which is a college textbook for a first course of elementary calculus. There, the principles of nonstandard analysis are presented axiomatically in a nice and elementary form (see the accompanying book [32] for the foundational aspects). Among the more recent works, there are the “gentle” introduction by W.C. Henson [26], R. Goldblatt's lectures on the hyperreals [25], and K.D. Stroyan's textbook [44].

Recently the authors investigated several different frameworks in algebra, topology, and set theory, that turn out to incorporate explicitly or implicitly the “hyper-methods”. These approaches show that nonstandard extensions naturally arise in several quite different contexts of mathematics. An interesting phenomenon is that some of those approaches lead in a straightforward manner to ultrafilter properties that are independent of the axioms of Zermelo-Fraenkel set theory ZFC.