Nonstandard analysis is an approach to doing calculus and mathematical analysis that uses infinitesimals in a consistent way (in contrast with early approaches that were inconsistent, and were replaced by limits). Infinitesimals, and other nonstandard objects, are constructed using techniques from model theory.
I'll explain the model-theoretic constructions used in nonstandard analysis, including filters, ultrafilters, ultraproducts, and ultrapowers, and how they're used to construct infinitesimals. I'll finish by presenting a 2003 paper of Kanovei and Shelah that gives a definable nonstandard model of the reals. Along the way, I'll cover other topics, possibly including