Foundations -- The strength of nonstandard analysis -- The virtue of simplicity -- Analysis of various practices of referring in classical or non standard mathematics -- Stratified analysis? -- ERNA at work -- The Sousa Pinto approach to nonstandard generalised functions -- Neutrices in more dimensions -- Number theory -- Nonstandard methods for additive and combinatorial number theory. A survey -- Nonstandard methods and the Erd?s-Turán conjecture -- Statistics, probability and measures -- Nonstandard likelihood ratio test in exponential families -- A finitary approach for the representation of the infinitesimal generator of a markovian semigroup -- On two recent applications of nonstandard analysis to the theory of financial markets -- Quantum Bernoulli experiments and quantum stochastic processes -- Applications of rich measure spaces formed from nonstandard models -- More on S-measures -- A Radon-Nikodým theorem for a vector-valued reference measure -- Differentiability of Loeb measures -- Differential systems and equations -- The power of Gâteaux differentiability -- Nonstandard Palais-Smale conditions -- Averaging for ordinary differential equations and functional differential equations -- Path-space measure for stochastic differential equation with a coefficient of polynomial growth -- Optimal control for Navier-Stokes equations -- Local-in-time existence of strong solutions of the n-dimensional Burgers equation via discretizations -- Infinitesimals and education -- Calculus with infinitesimals -- Pre-University Analysis.

Nonstandard Analysis enhances mathematical reasoning by introducing new ways of expression and deduction. Distinguishing between standard and nonstandard mathematical objects, its inventor, the eminent mathematician Abraham Robinson, settled in 1961 the centuries-old problem of how to use infinitesimals correctly in analysis. Having also worked as an engineer, he saw not only that his method greatly simplified mathematically proving and teaching, but also served as a powerful tool in modelling, analyzing and solving problems in the applied sciences, among others by effective rescaling and by infinitesimal discretizations. This book reflects the progress made in the forty years since the appearance of Robinson’s revolutionary book Nonstandard Analysis: in the foundations of mathematics and logic, number theory, statistics and probability, in ordinary, partial and stochastic differential equations and in education. The contributions are clear and essentially self-contained.