The rapid development of artificial intelligence (AI) requires one to speed up the development of the domain-specific hardware specifically designed for AI applications. The neuromorphic computing architecture consisting of synapses and neurons, which is inspired by the integrated storage and parallel processing of human brain, can effectively reduce the energy consumption of artificial intelligence in computing work. Memory components have shown great application value in the hardware implementation of neuromorphic computing. Compared with traditional devices, the memristors used to construct synapses and neurons can greatly reduce computing energy consumption. However, in neural networks based on memristors, updating and reading operations have system energy loss caused by voltage and current of memristors. As a derivative of memristor, memcapacitor is considered as a potential device to realize a low energy consumption neural network, which has attracted wide attention from academia and industry. Here, we review the latest advances in physical/simulated memcapacitors and their applications in neuromorphic computation, including the current principle and characteristics of physical/simulated memcapacitor, representative synapses, neurons and neuromorphic computing architecture based on memcapacitors. We also provide a forward-looking perspective on the opportunities and challenges of neuromorphic computation based on memcapacitors.