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In these lecture notes we present an introduction to non-standard analysisespecially written for the community of mathematicians, physicists andengineers who do research on J. F. Colombeau' theory of new generalizedfunctions and its applications. The main purpose of our non-standard approachto Colombeau' theory is the improvement of the properties of the scalars of thevarieties of spaces of generalized functions: in our non-standard approach thesets of scalars of the functional spaces always form algebraically closednon-archimedean Cantor complete fields. In contrast, the scalars of thefunctional spaces in Colombeau's theory are rings with zero divisors. Theimprovement of the scalars leads to other improvements and simplifications ofColombeau's theory such as reducing the number of quantifiers and possibilitiesfor an axiomatization of the theory. Some of the algebras we construct in thesenotes have already counterparts in Colombeau's theory, other seems to bewithout counterpart. We present applications of the theory to PDE andmathematical physics. Although our approach is directed mostly to Colombeau'scommunity, the readers who are already familiar with non-standard methods mightalso find a short and comfortable way to learn about Colombeau's theory: a newbranch of functional analysis which naturally generalizes the Schwartz theoryof distributions with numerous applications to partial differential equations,differential geometry, relativity theory and other areas of mathematics andphysics.

Lecture Notes: Non-Standard Approach to J.F. Colombeau's Theory of Generalized Function: University of Vienna, Austria, May 2006