  A geometric grid class consists of those permutations that can be drawn on a
specified set of line segments of slope \pm1 arranged in a rectangular pattern
governed by a matrix. Using a mixture of geometric and language theoretic
methods, we prove that such classes are specified by finite sets of forbidden
permutations, are partially well ordered, and have rational generating
functions. Furthermore, we show that these properties are inherited by the
subclasses (under permutation involvement) of such classes, and establish the
basic lattice theoretic properties of the collection of all such subclasses.
