Inherent memory in memristor resistance represents digital or analog information in both memory and computer engineering. After Professor Chua introduced the Memristor, a non-volatile device, substantial research has been dedicated to unveiling its potential. Memristors, two-terminal devices with built-in memory, exhibit the ability to operate effectively in parallel computing, simultaneously handling storage and computational tasks. This unique capability is employed by modulating their conductance through the manipulation of oxygen vacancies. Notably, Memristors can also emulate the behavior of brain cells, contributing to the remarkable progress in artificial intelligence, which has left a significant impact on both academic and industrial realms. By emulating brain-like synaptic functions, memristors bridge the gap between traditional computing and neuromorphic architectures, enabling efficient processing and learning. This capability has sparked widespread interest across academic and industrial sectors, driving the development of systems that combine compactness, energy efficiency, and high computational performance. In this book chapter we, focused our efforts on a detailed examination of a vertical nanowire crossbar array of Hafnium Oxide (HfO2) as it offers a range of exceptional characteristics, including high stability, a wide band gap, outstanding electrical properties, a high dielectric permittivity (ranging from ε = 15 to 27), robust endurance, and impressive performance. These attributes contribute to the robustness and reliability of HfO2-based memristors, making them ideal for practical applications. Furthermore, their ability to operate with low switching currents in the nanoampere range enhances energy efficiency, a critical factor for modern computing systems. The crossbar architecture of memristor arrays offers remarkable advantages, including dense packing and individual accessibility of nanoscale devices. These features enable the creation of highly scalable, energy-efficient systems capable of handling complex tasks. The nonlinear switching behavior exhibited by memristors enhances their functionality, supporting diverse computational requirements while minimizing power consumption. The integration of such arrays within neuromorphic systems holds immense promise for advancing artificial intelligence and computing paradigms. The potential of memristor-based systems extends beyond conventional applications, encompassing a wide range of fields such as data storage, signal processing, and machine learning. The unique combination of memory and computation within a single device, coupled with the exceptional material properties of HfO2, positions memristors as a cornerstone of future technological innovations. Research into these systems continues to reveal new possibilities, emphasizing their critical role in shaping the next generation of energy-efficient, high-performance computing solutions.