Memristors mimic synaptic functions in advanced electronics and image sensors, thereby enabling brain-inspired neuromorphic computing to overcome the limitations of the von Neumann architecture. As computing operations based on von Neumann hardware rely on continuous memory transport between processing units and memory, fundamental limitations arise in terms of power consumption and integration density. In biological synapses, chemical stimulation induces information transfer from the pre- to the post-neuron. The memristor operates as resistive random-access memory (RRAM) and is incorporated into the hardware for neuromorphic computing. Hardware composed of synaptic memristor arrays is expected to lead to further breakthroughs owing to their biomimetic in-memory processing capabilities, low power consumption, and amenability to integration; these aspects satisfy the upcoming demands of artificial intelligence for higher computational loads. Among the tremendous efforts toward achieving human-brain-like electronics, layered 2D materials have demonstrated significant potential owing to their outstanding electronic and physical properties, facile integration with other materials, and low-power computing. This review discusses the memristive characteristics of various 2D materials (heterostructures, defect-engineered materials, and alloy materials) used in neuromorphic computing for image segregation or pattern recognition. Neuromorphic computing, the most powerful artificial networks for complicated image processing and recognition, represent a breakthrough in artificial intelligence owing to their enhanced performance and lower power consumption compared with von Neumann architectures. A hardware-implemented CNN with weight control based on synaptic memristor arrays is expected to be a promising candidate for future electronics in society, offering a solution based on non-von Neumann hardware. This emerging paradigm changes the computing algorithm using entirely hardware-connected edge computing and deep neural networks.