  We investigate a generalization of stacks that we call
$\mathcal{C}$-machines. We show how this viewpoint rapidly leads to functional
equations for the classes of permutations that $\mathcal{C}$-machines generate,
and how these systems of functional equations can frequently be solved by
either the kernel method or, much more easily, by guessing and checking.
General results about the rationality, algebraicity, and the existence of
Wilfian formulas for some classes generated by $\mathcal{C}$-machines are
given. We also draw attention to some relatively small permutation classes
which, although we can generate thousands of terms of their enumerations, seem
to not have D-finite generating functions.
