Provided are resistive random access memory (ReRAM) cells, each having three or more resistive states and being capable of storing multiple bits of data, as well as methods of fabricating and operating such ReRAM cells. Such ReRAM cells or, more specifically, their resistive switching layer have wide range of resistive states and are capable of being very conductive (e.g., about 1 kOhm) in one state and very resistive (e.g., about 1 MOhm) in another state. In some embodiments, a resistance ratio between resistive states may be between 10 and 1,000 even up to 10,000. The resistive switching layers also allow establishing stable and distinct intermediate resistive states that may be assigned different data values. These layers may be configured to switching between their resistive states using fewer programming pulses than conventional systems by using specific materials, switching pluses, and resistive state threshold.