Nonstandard analysis is a set of techniques whose central idea is to enlarge a given mathematical structure by adding certain “ideal” elements in such a way that the enlarged structure maintains the same “logical” properties as the original structure.  Nonstandard analysis has found applications in nearly every area of mathematics.  In this talk, we will explain how this technique works by giving some simple proofs of important theorems from Ramsey theory, which is a branch of combinatorics.  These applications involve a relatively recent idea, namely the notion of an iterated nonstandard extension.