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We present our extension of Allard's celebrated rectifiability theorem to the setting of varifolds with locally bounded anisotropic first variation. We identify a necessary and sufficient condition on the integrand for its validity and we discuss the connections of this condition to Almgren's ellipticity. We apply this result to the set-theoretic anisotropic Plateau problem, obtaining solutions to three different formulations: one ntroduced by Reifenberg, one proposed by Harrison and Pugh and another one studied by David. Moreover, we apply the rectifiability theorem to prove an anisotropic counterpart of Allard's compactness result for integral varifolds Some of the presented theorems are joint works with De Lellis, De Philippis, Ghiraldin and Kolasiński.