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Hybrid circuits consisting of a CMOS chip and add-on nanowire crossbar have gained much attention recently since this concept can compensate for the limited functionality of nanodevices fabricated with the reproducibility necessary for integrated circuits. In this concept, two perpendicular layers of nanowire of crossbar are connected with two-terminal latching switches formed at every cross point. In the 'CMOL' variety of hybrid circuits, each individual nanowire can be addressed individually from the CMOS subsystem through an interconnect pin. As a result, each latching switch can be turned on or off independently. The main advantage of such systems comes from the unprecedented density of the nanowire crossbar. According to our estimates, CrossNets-mixed-signal neuromorphic architectures, based on CMOL-may outperform the cerebral cortex in terms of areal density. This fact encouraged us to explore these CrossNets. This dissertation describes CrossNet-based Multi-Layer Perceptrons (MLP) with discrete and defective synapses, and in particular various options of their supervised training as pattern classifiers. The simplest training option here is "weight import"; after training a precursor network, synaptic weights are copied to the discrete synapses of CrossNets. This rule has been applied successfully to a large-scale benchmark task MNIST. However, weight import training with a precursor network may become impracticably slow when the required network size is too large. In order to train such large networks, weights should be computed inside a CrossNet without a precursor network. Our idea of such in-situ training algorithms is based on the stochastic multiplication, implemented with Cross-Net synapses. We have shown that the in-situ algorithm can be used to train CrossNet-based MLP to perform handwritten digit recognition. Moreover, the trained CrossNets have demonstrated high defect tolerance. We believe our research will help to design, in future, more complex (hierarchical/modular) systems working similarly to our brain.