In recent years, α-MoO₃ has received extensive attention in the research of memristor devices. The variation of valence of molybdenum will lead the resistivity to change, and the unique layer structure is beneficial to the implantation of donor ion into free space to adjust the conductance, so that it has a great influence on the study of synaptic transistors. This paper mainly summarize the properties of α-MoO₃, the method and characteristics of large-scale two-dimensional α-MoO₃ and analyze the recent progress of in-memory computing based on α-MoO₃. Primarily, this paper introduces crystal structures, band structure and defect state of α-MoO₃. The synthesis methods of large-area α-MoO₃ are compared with each other, including the one-step method to directly obtain α-MoO₃ nanosheets, and the combined post-annealing process of magnetron sputtering or atomic layer deposition to prepare the thin α-MoO₃ films. In the one-step synthesis method we conclude that the chamber pressure influences the ratio of MoO₃ to MoO_3–x, and the growth temperature affects the ratio of α-MoO₃ to β-MoO₃. That is to say, the phase composition of molybdenum trioxide, the concentration of precursors has an important influence on the film size. The advantages of α-MoO₃ synthesis by different methods in memory computing applications are discussed in detail. And then, this paper summarizes the device performance of α-MoO₃ in memristor and the application progress of α-MoO₃-based neuromorphic devices, and analyzes the performance of α-MoO₃-based resistive random access memory such as switching ratio, endurance, and stability in detail. The synaptic functions of different structural device units are extensively studied, and various typical synapse functions are realized such as short-term plasticity, long-term plasticity, paired pulse facilitation, etc. It shows the excellent characteristics of low energy consumption in the simulation of synaptic plasticity. The use of short-term memory and long-term memory modes of the device can realize the functions of image memory and preprocessing. Through the memristor array combined with the back-propagation network, the high-precision handwriting recognition can be realized. Finally, combining the recent research progress of α-MoO₃, its opportunities and challenges in the field of memory computing are prospected.