Neuromorphic computing, which mimics the energy-efficient parallel processing capabilities of the human brain, has emerged as an alternative to traditional von Neumann architectures that struggle with high power consumption in the era of artificial intelligence (AI). Despite the potential of Si-based neuromorphic chips, they often face fundamental limitations in integration density and biological compatibility, necessitating the development of next-generation devices that can better emulate the ionic signaling of biological systems. This review provides a comprehensive analysis of the recent research trends in artificial synapses and neurons based on organic electrochemical transistors (OECTs), highlighting their unique ability to achieve high transconductance and mixed ionic-electronic conduction at ultra-low operating voltages. We discuss how OECTs successfully replicate diverse synaptic plasticities and complex neuronal spiking behaviors through advanced material engineering and structural optimizations such as vertical architectures. Furthermore, this review discusses the implementation of high-order neural functions, including associative learning and logic operations, which are facilitated by the inherent electrochemical dynamics of organic semiconductors. Finally, overcoming current challenges in reliability and scalability will establish OECTs as a pivotal platform for low-power neuromorphic hardware and bio-integrated electronics.