  In this paper we solve a long standing problem about the multivariable Rubio
de Francia extrapolation theorem for the multilinear Muckenhoupt classes
$A_{\vec{p}}$, which were extensively studied by Lerner et al. and which are
the natural ones for the class of multilinear Calder\'on-Zygmund operators.
Furthermore, we go beyond the classes $A_{\vec{p}}$ and extrapolate within the
classes $A_{\vec{p},\vec{r}}$ which appear naturally associated to the weighted
norm inequalities for multilinear sparse forms which control fundamental
operators such as the bilinear Hilbert transform. We give several applications
which can be easily obtained using extrapolation. First, for the bilinear
Hilbert transform one can extrapolate from the recent result of Culiuc et al.
who considered the Banach range and extend the estimates to the quasi-Banach
range. As a direct consequence, we obtain weighted vector-valued inequalities
reproving some of the results by Benea and Muscalu. We also extend recent
results of Carando et al. on Marcinkiewicz-Zygmund estimates for multilinear
Calder\'on-Zygmund operators. Finally, our last application gives new weighted
estimates for the commutators of multilinear Calder\'on-Zygmund operators and
for the bilinear Hilbert transform with BMO functions using ideas from B\'enyi
et al.

Classical Analysis and ODEs