  A rational function of degree at least two with coefficients in an
algebraically closed field is post-critically finite (PCF) if all of its
critical points have finite forward orbit under iteration. We show that the
collection of PCF rational functions is a set of bounded height in the moduli
space of rational functions over the complex numbers, once the well-understood
family known as flexible Lattes maps is excluded. As a consequence, there are
only finitely many conjugacy classes of non-Lattes PCF rational maps of a given
degree defined over any given number field. The key ingredient of the proof is
a non-archimedean version of Fatou's classical result that every attracting
cycle of a rational function over the complex numbers attracts a critical
point.
