In this paper, we study Z(2)Z(4)-additive cyclic codes. These codes are identified as Z(4)[x]-submodules of the ring R-r,R-s = Z(2)[x]/< x(r) - 1 > x Z(4) [x]/< x(s)-1 >. The algebraic structure of this family of codes is studied and a set of generator polynomials for this family as a Z(4)[x]-submodule of the ring R-r,R-s is determined. We show that the duals of Z(2)Z(4)-additive cyclic codes are also cyclic. We also present an infinite family of Maximum Distance separable with respect to the singleton bound codes. Finally, we obtain a number of binary linear codes with optimal parameters from the Z(2)Z(4)-additive cyclic codes.