  Ultrafilters are a tool, originating in mathematical logic and general
topology, that has steadily found more and more uses in multiple areas of
mathematics, such as combinatorics, dynamics, and algebra, among others. The
purpose of this article is to introduce ultrafilters in a friendly manner and
present some applications to the branch of combinatorics known as Ramsey
theory, culminating with a new ultrafilter-based proof of van der Waerden's
theorem.
