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This tutorial describes challenges and possible avenues for the implementation of the components of a solid-state system, which emulates a biological brain. The tutorial is devoted mostly to a charge-based (i.e. electric controlled) implementation using transition metal oxide materials, which exhibit unique properties that emulate key functionalities needed for this application. In Sec. I, we compare the main differences between a conventional computational machine, based on the Turing-von Neumann paradigm, and a neuromorphic machine, which tries to emulate important functionalities of a biological brain. We also describe the main electrical properties of biological systems, which would be useful to implement in a charge-based system. In Sec. II, we describe the main components of a possible solid-state implementation. In Sec. III, we describe a variety of Resistive Switching phenomena, which may serve as the functional basis for the implementation of key devices for neuromorphic computing. In Sec. IV, we describe why transition metal oxides are promising materials for future neuromorphic machines. Theoretical models describing different resistive switching mechanisms are discussed in Sec. V, while existing implementations are described in Sec. VI. Section VII presents applications to practical problems. We list in Sec. VIII important basic research challenges and open issues. We discuss issues related to specific implementations, novel materials, devices, and phenomena. The development of reliable, fault tolerant, energy efficient devices, their scaling, and integration into a neuromorphic computer may bring us closer to the development of a machine that rivals the brain.