Hybrid CMOS/nanoelectronic circuits, combining CMOS chips with simple nanoelectronic crossbar add-ons, may extend the exponential Moore-Law progress of microelectronics beyond the 10-nm frontier. This paper reviews the development of neuromorphic networks ("CrossNets") based on this prospective technology. In these networks, the neural cell bodies ("somas") are implemented in the CMOS subsystem, crossbar nanowires are used as axons and dendrites, while two-terminal crosspoint devices are used as elementary synapses. Extensive analysis and simulations have shown that such networks may perform virtually all information processing tasks demonstrated with software-implemented neural networks, with much higher performance. Estimates show that CrossNets may eventually overcome bio-cortical circuits in density, at comparable connectivity, while operating 4 to 6 orders of magnitude faster, at manageable power dissipation.