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The first author is supported by NSF Grant DMS-1362425 and research funds from the Dean of the College of Arts & Sciences, the University of Alabama. The second author is supported by the Simons Foundation. We establish the boundedness of the multilinear Calderon{Zygmund operators from a product of weighted Hardy spaces into a weighted Hardy or Lebesgue space. Our results generalize to the weighted setting results obtained by Grafakos and Kalton [18] and recent work by the third author, Grafakos, Nakamura, and Sawano [20]. As part of our proof we provide a finite atomic decomposition theorem for weighted Hardy spaces, which is interesting in its own right. As a consequence of our weighted results, we prove the corresponding estimates on variable Hardy spaces. Our main tool is a multilinear extrapolation theorem that generalizes a result of the first author and Naibo [10].