Brain-inspired neuromorphic electronics have been extensively studied as systems for wearable devices, neuroprostheses, and soft machines, offering solutions to the limitations of conventional von Neumann computing systems and enabling efficient information processing. Among these, synaptic transistors with vertical structures are gaining significant attention as promising candidates for flexible neuromorphic electronics, owing to their unique structural features, such as ultrashort channel lengths and vertical carrier transport, which provide superior performance, mechanical flexibility, and high-density integration. Vertical synaptic transistors (VSTs) not only combine the functionalities of information processing, memory, and sensing/responding within a single device but also enable the realization of diverse synaptic properties, effectively mimicking the information processing and sensory capabilities of biological nervous systems. Achieving both mechanical flexibility and excellent electrical performance in VSTs necessitates a strong focus on the active layer, prompting extensive research into various flexible semiconducting materials. This review explores the diverse range of flexible semiconducting materials employed in VSTs and their fundamental operating mechanisms. Additionally, it highlights recent advancements in VSTs and systems developed to replicate the functionalities of biological nervous systems.