<p>The remarkable computational power of the human brain has long inspired the development of intelligent systems. As described in the foundational McCulloch-Pitts model, this power arises from networks of interconnected neurons that communicate through axons, synapses, and dendrites, with neurons modelled as binary switches activated by weighted inputs. Over time, this concept evolved to include variable synaptic strengths and multilayered architectures, paving the way for modern artificial neural networks (ANNs) and their specialized forms such as convolutional neural networks (CNNs), recurrent neural networks (RNNs), spiking neural networks (SNNs), and the transformer neural networks that now power many advanced AI systems. These models have shown exceptional performance, especially in parallel matrix-vector computations. However, their scalability is highly hindered by the von Neumann bottleneck inherent in traditional Complementary Metal-Oxide-Semiconductor (CMOS)-based architectures, where the physical separation between memory and processing units results in latency and energy inefficiencies particularly under the demands of large dynamic datasets. To address this, researchers are embracing computing-in-memory (CIM) paradigms, which aim to eliminate this separation by executing computations directly within memory arrays. Memristor crossbar architectures are central to this transition, leveraging analog parallel processing based on physical laws to enable efficient matrix operations. Acting as tuneable synapses, memristors support high-density integration and mimic biological learning mechanisms. This paper traces the evolution from early neuron models to cutting-edge neural architectures and highlights how memristor-based CIM systems can bridge the gap between biological inspiration and hardware efficiency, opening new frontiers for data-intensive, brain-like computing. For the same, recent strategies to map large-scale neural networks onto limited CIM hardware and advanced training methods designed to cope with the non-idealities of memristive devices are presented. A special emphasis is placed on compiler-driven mapping, quantization-aware and device-aware training, and negative-feedback stabilization as essential enablers for robust artificial intelligence (AI) hardware. Finally, Crucial challenges and research trends are also outlined, including hybrid precision architectures, bio-inspired learning, and multifunctional CIM platforms that extend beyond conventional AI tasks toward advanced biomedical tools, intelligent sensing systems, and scalable industrial applications.</p>