The unstructured data such as visual information, natural language, and human behaviors opens up a wide array of opportunities in the field of artificial intelligence (AI). The memory-centric neuromorphic computing (MNC) has been proposed for the efficient processing of unstructured data, bypassing the von Neumann bottleneck of current computing architecture. The development of MNC would provide massively parallel processing of unstructured data, realizing the cognitive AI in edge and wearable systems. In this review, recent advances in memory-centric neuromorphic devices are discussed in terms of emerging nonvolatile memories, volatile switches, synaptic plasticity, neuronal models, and memristive neural network.