A variable Lebesgue space is a generalization of the classical $L^p$ spaces with the constant exponent $p$ replaced by a measurable function $p(\cdot)$. Extensive work has been done on these spaces in the past 20 years, both for their intrinsic interest and for their applications to PDEs and the calculus of variations. More recently there has been interest in generalizing the theory of weighted norm inequalities due to Muckenhoupt and others to this setting. In this talk we will discuss our work with C.J. Neugebauer, A. Fiorenza and D. Wang in this area. We will give necessary and sufficient conditions for the Hardy-Littlewood maximal operator to be bounded, and then show how this can be used to extend Rubio de Francia extrapolation to this setting. Extrapolation then lets us prove that a large number of operators are bounded on weighted $L^{p(\cdot)}$ spaces.