A range of material systems exist in which nanoscale ionic transport and redox reactions provide the essential mechanisms for memristive switching. One class relies on mobile cations, which are easily created by electrochemical oxidation of the corresponding electrode metal, transported in the insulating layer, and reduced at the inert counterelectrode. These devices are termed electrochemical metallization (ECM) memories, also called conductive bridge random access memories. The memristive characteristics of the ECM cells provide opportunities for circuit design and computational concepts that go beyond those in traditional complementary metal oxide semiconductor (CMOS) technology. Passive memory arrays open up paths toward ultradense and 3D stackable memory and logic gate arrays. Furthermore, the multivalued conductance characteristics allow for potential exploitation of the cells as synapses in neuromorphic circuits in future energy efficient high-performance computer architectures. Despite exciting results obtained in recent years, many challenges have to be met before these physical effects can be turned into competitive industrial technology. Here, we briefly review the basic working principle, the different possible and potential material combinations, and the fundamental electrochemical processes in ECM cells and their implications for device operations. The prospects of ECM-based resistive random access memory as an emerging memory technology are also reviewed in terms of switching speed and scalability.