What are the most important applications *outside* of mathematics of each of the major fields of mathematics? For concreteness, let's divide up mathematics according to [arxiv mathematics categories](https://arxiv.org/archive/math), e.g. math.AT, math.QA, math.CO, etc.

This is a community-wiki question, so please edit and improve pre-existing answers: let's keep it to a **single answer for each subject area.**

(This is inspired by Terry Tao's [recent post](http://terrytao.wordpress.com/2009/10/25/applications-oriented-periodic-table/) about a periodic table of the elements listing commercial applications. He suggested it might be fun to have such a summary for either the MSC top-level subjects or the arxiv subjects.)

I'd like to propose that for areas in which the applications are either numerous, non-obvious, or generally worthy of discussion, someone volunteers to open up a new question specifically about that subject area, and takes care of providing a summary here of the best answers produced there.