Summary: The authors discuss a class of multilinear singular integrals and obtain their boundedness from the weighted Hardy spaces \(H^1_\omega(\mathbb{R}^n)\) to the weighted Lebesgue space \(L^1_\omega(\mathbb{R}^n)\) for \(\omega\in A_1(\mathbb{R}^n)\) (the class of Muckenhoupt’s weights) and from the weighted Herz-type Hardy space \(H\dot K_p(\omega_1,\omega_2; \mathbb{R}^n)\) (or \(HK_p(\omega_1,\omega_2; \mathbb{R}^n)\)) to the weighted Herz space \(\dot K_p(\omega_1, \omega_2; \mathbb{R}^n)\) (or \(K_p(\omega_1, \omega_2; \mathbb{R}^n)\)) for any \(p\in (1,\infty)\) and \(\omega_1,\omega_2\in A_1(\mathbb{R}^n)\).