TITLE: Fixed point theorems
QUESTION [42 upvotes]: It is surprising that fixed point theorems (FPTs) appear in so many different contexts throughout Mathematics: Applying Kakutani's FPT earned Nash a Nobel prize; I am aware of some uses in logic; and of course everyone should know Picard's Theorem in ODEs. There are also results about local and global structure OF the fixed points themselves, and quite some famous conjectures (also labeled FPT for the purpose of this question).
Many results are so far removed from my field that I am sure there are plenty of FPTs out there that I have never encountered. I know of several, and will post later if you do not beat me to them :)
Community wiki rules apply. One FPT per answer, preferably with an inspiring list of interesting applications.

REPLY [8 votes]: I forgot who proved it, but the statement is nice and very easy to prove: A function $f:X\to X$ is fixed point free if and only if there is a partition of $X$ into three subsets s.t. $f$ maps each of the three subsets into the union of the other two.  An immediate application is that if $f$ is fixed point free on a set $X$ then so is its continuous extension to a function on $\beta X$.