Neuromorphic engineering is rapidly developing as an approach to mimicking processes in brains using artificial memristors, devices that change conductivity in response to the electrical field (resistive switching effect). Memristor-based neuromorphic systems can overcome the existing problems of slow and energy-inefficient computing that conventional processors face. In the Introduction, the basic principles of memristor operation and its applications are given. The history of switching in sandwich structures and granular metals is reviewed in the Historical Overview. Particular attention is paid to the fundamental articles from the pre-memristor era (the 1960s-70s), which demonstrated the first evidence of resistive switching and predicted the filamentary mechanism of switching. Multi-dimensionality in neuromorphic systems: Despite the powerful computational abilities of traditional memristor arrays, they cannot repeat many organizational characteristics of biological neural networks, i.e., their multi-dimensionality. This part reviews the unconventional nanowire- and nanoparticle-based neuromorphic systems that demonstrate incredible potential for use in reservoir computing due to the unique spiking change in conductance similar to firing in neurons. Liquid-based neuromorphic devices: The transition of neuromorphic systems from solid to liquid state broadens the possibilities for mimicking biological processes. In this section, ionic current memristors are reviewed and, the working principles of which bring us closer to the mechanisms of information transmittance in real synapses. Nanofluids: A novel direction in neuromorphic engineering linked to the application of nanofluids for the formation of reconfigurable nanoparticle networks with memristive properties is given in this section. The Conclusion t summarizes the bullet points of the Review and provides an outlook on the future of liquid-state neuromorphic systems.