In this talk we present some recent results obtained in a joint work with K. Li and J. M. Martell, about a multivariable Rubio de Francia extrapolation theorem for multilinear Muckenhoupt classes $A_{\vec{p}}$, and also some extensions to more general classes of weights.To illustrate the power of extrapolation methods we will present some applications of the beforementioned results and some mixed weak-type weighted estimates obtained in a joint work with K. Li and C. P\'erez.