On the Generalized Marcinkiewicz Integral Operators with Rough Kernels A class of generalized Marcinkiewicz integral operators is introduced, and, under rather weak conditions on the integral kernels, the boundedness of such operators on $L^p$ and Triebel--Lizorkin spaces is established. Keywords: Marcinkiewicz integral, Littlewood--Paley theory, Triebel--Lizorkin space, rough kernel, product domainCategories:42B20, , , , , 42B25, 42B30, 42B99