  We extend the theory of Rubio de Francia extrapolation, including
off-diagonal, limited range, and $A_{\infty}$ extrapolation, to the weighted
variable Lebesgue spaces. As a consequence we are able to show that a number of
different operators from harmonic analysis are bounded on these spaces. The
proofs of our extrapolation results are developed in a way that outlines a
general approach to proving extrapolation theorems on other Banach function
spaces.
