Artificial neural networks are intended to solve complex machine-learning tasks by using massive parallel data processing in a similar way as in biological neural systems. A recent approach to mimic biology is to emulate the basic processing elements directly in hardware. ReRAM devices are considered key elements to realize highly scalable, and low-power neuromorphic systems consist of using a hybrid analog–digital circuit approach. The specific properties of ReRAM devices that make those devices highly useful as artificial synapse are highlighted in detail. Especially, the multilevel capability of ReRAM devices enables the implementation of learning rules such as spike-timing-dependent plasticity (STDP). The scaling perspectives of ReRAM-based neuromorphic architectures are elaborated on, revealing a scaling potential below 10 nm. Finally, several neuromorphic applications using ReRAM architectures are reviewed and an evaluation of the future perspectives is given.