Loeb Measures in Practice: Recent Advances This expanded version of the 1997 European Mathematical Society Lectures given by the author in Helsinki begins with a self contained introduction to nonstandard analysis (NSA) and the construction of Loeb Measures which are rich measures discovered in 1975 by Peter Loeb using techniques from NSA. Subsequent chapters sketch a range of recent applications of Loeb measures due to the author and his collaborators in such diverse fields as (stochastic) fluid mechanics stochastic calculus of variations ( Malliavin calculus) and the mathematical finance theory. The exposition is designed for a general audience and no previous knowledge of either NSA or the various fields of applications is assumed. Loeb Measures: Introduction Nonstandard Analysis Construction of Loeb Measures Loeb Integration Theory Elementary Applications. Stochastic Fluid Mechanics: Introduction Solution of the Deterministic Navier Stokes Equations Solution of the Stochastic Navier Stokes Equations Stochastic Euler Equations Statistical Solutions Attractors for the Navier Stokes Equations Measure Attractors for Stochastic Navier Stokes Equations Stochastic Attractors for Navier Stokes Equations Attractors for the 3 dimensional Stochastic Navier Stokes Equations. Stochastic Calculus of Variations: Introduction Flat Integral Representation of Wiener Measure The Wiener Sphere Brownian Motion on the Wiener Sphere and the Infinite Dimensional Ornstein Uhlenbeck Process Malliavin Calculus. Mathematical Finance Theory: Introduction The Cox Ross Rubinstein Models Options and Contingent Claims The Black Scholes Model... The complete table of contents can be found on the Internet: http: www.springer.de.