Brain-inspired computing has been the subject of an intense research in the last decades, with the aim of recreating some of the cognitive computing functions of the human brain in silicon-based hardware. In this frame, resistive switching memories (RRAM) and other emerging memory technologies are extremely promising as they offer memory and plasticity with high scaling capability, thus enabling the integration of a high density of synapses and neurons. This chapter summarizes the status and challenges of RRAM-based neuromorphic engineering. RRAM synapses are described within the frame of various neural network architectures, such as artificial neural networks (ANNs) and spiking neural networks (SNNs). First, ANNs with deep neural network architectures are described in terms of their operation during inference and learning, referring to the typical backpropagation scheme for supervised training. The challenges for high-density, high-functionality ANNs for computer vision with RRAM synapses are addressed.RRAM circuits enabling spike-timing dependent plasticity (STDP) and their use for unsupervised learning in feed-forward and recurrent networks are then presented. A hardware SNN for unsupervised learning by STDP in RRAM synapses is illustrated, demonstrating learning of static and dynamic patterns with both binary and gray-scale values. Finally, an outlook on the prospects of RRAM for future cognitive computing is given.