ZnO thin films were deposited on heavily doped silicon wafer by DC magnetron sputtering and the Cu electrodes were evaporated on ZnO/ n+-Si by electric beam evaporation to get Cu/ZnO/n+-Si resistive random access memory (ReRAM). The forming, reset and set processes of the devices were investigated using filamentary model. The effects of film thickness on the crystalline structure of the ZnO thin films and the resistive switching characteristics of the fabricated devices were investigated. The diffraction peak intensity and crystal size increased with increasing film thickness, which shows better crystallization. Cu/ZnO/n+-Si structured device exhibits reversible and steady unipolar resistive switching behaviors. The film thickness had great effect on the forming process of the prepared devices, while the values of Vset increased and Vreset varied little with increasing the film thickness.