The paper presents an analysis of current state and perspectives of high-performance com- puting based on the principles of information storage and processing in biological neural networks, which are enabled by the new micro- and nanoelectronics component base. Its key element is the memristor (associated with a nonlinear resistor with memory or Resistive Random Access Memory (RRAM) device), which can be implemented on the basis of different materials and nanostruc- tures compatible with the complementary metal-oxide-semiconductor (CMOS) process and allows computing in memory. This computing paradigm is naturally implemented in neuromorphic sys- tems using the crossbar architecture for vector-matrix multiplication, in which memristors act as synaptic weights – plastic connections between artificial neurons in fully connected neural network architectures. The general approaches to the development and creation of a new component base based on the CMOS-integrated RRAM technology, development of artificial neural networks and neuroprocessors using memristive crossbar arrays as computational cores and scalable multi-core architectures for implementing both formal and spiking neural network algorithms are discussed. Technical solutions are described that enable hardware implementation of memristive crossbars of sufficient size, as well as solutions that compensate for some of the deficiencies or fundamental limitations inherent in emerging memristor technology. The performance and energy efficiency are analyzed for the reported prototypes of such neuromorphic systems, and a significant (orders of magnitude) gain in these parameters is highlighted compared to the computing systems based on traditional component base (including neuromorphic ones). Technological maturation of a new component base and creation of memristor-based neuromorphic computing systems will not only provide timely diversification of hardware for the continuous development and mass implemen- tation of artificial intelligence technologies but will also enable setting the tasks of a completely new level in creating hybrid intelligence based on the symbiosis of artificial and biological neural networks. Among these tasks are the primary ones of developing brain-like self-learning spiking neural networks and adaptive neurointerfaces based on memristors, which are also discussed in the paper.