Modern artificial intelligence (AI) systems, based on von Neumann architecture and classical neural networks, have a number of fundamental limitations in comparison with the mammalian brain. In this article we discuss these limitations and ways to mitigate them. Next, we present an overview of currently available neuromorphic AI projects in which these limitations are overcome by bringing some brain features into the functioning and organization of computing systems (TrueNorth, Loihi, Tianjic, SpiNNaker, BrainScaleS, NeuronFlow, DYNAP, Akida, Mythic). Also, we present the principle of classifying neuromorphic AI systems by the brain features they use: connectionism, parallelism, asynchrony, impulse nature of information transfer, on-device-learning, local learning, sparsity, analog, and in-memory computing. In addition to reviewing new architectural approaches used by neuromorphic devices based on existing silicon microelectronics technologies, we also discuss the prospects for using a new memristor element base. Examples of recent advances in the use of memristors in neuromorphic applications are also given.