Filters provide notions of both convergence and of size. We shall see how filters can be used to characterize a wide range of modes of convergence and then introduce the notion of an ultrafilter, which is a filter that is maximal under the natural notion of refinement for filters. A degenerate class of ultrafilters correspond to natural notions of convergence but others are mysterious objects which in a precise sense cannot be completely described.
Nevertheless the nondegenerate ultrafilters, which are called nonprincipal, give rise to surprising constructions in a variety of mathematical contexts. Ultrafilters are used to prove that every Boolean algebra is isomorphic to a field of sets, to construct the Stone-Chech compactification of a topological space, and to prove Hindman's theorem, a statement in combinatorial number theory.
An algebraic construction based on ultrafilters, called ultrapowers, can efficiently generate structures very similar to the natural numbers or to the real numbers but which have infinite or infinitesimal elements. This is just a small sampling of the many uses to which the mysterious objects known as ultrapowers can be put!