Solid electrolytes such as Agy(GexSe 1-x)1-y allow rapid diffusion of metal ions and this makes them suitable for memory applications. The switching mechanism in these materials is based on cation migration from an oxidizable electrode (e.g., Ag) under positive bias and the reduction of the metal ions at the counter electrode (e.g., Pt). A metallic connection forms between the electrodes which is stable when the voltage is switched off but can be dissolved when the voltage polarity is reversed. We present our results on resistive switching in Ag/Ag-Ge-Se/Pt cells which can show a resistance ratio of more than 5 orders of magnitude. The ON resistance depends on the write current which allows for multi-bit data storage. The leakage current in the high resistance state can significantly be reduced by introducing an oxide layer within the chalcogenide film. Then, a current as low as 1 nA is sufficient to switch these cells from a high to a low resistance state demonstrating the possibility of extremely low power consumption.