Neuromorphic engineering, inspired by the biological brain, is an emerging technique that uses electronic components to imitate the characteristics of biological neurons. Neuromorphic electronics provide several advantages, including learning capability, low power consumption, and rapid response, which offer a possible solution to improve the performance of electronic systems. Polymer-based field-effect transistors (FETs) with non-volatile memory characteristics have been considered promising devices for neuromorphic computing due to their unique electronic properties, excellent mechanical properties, solution processability, and biocompatibility. This review introduces the recent progress of polymer-based synaptic transistors from non-volatile memory mechanisms to artificial synaptic functions. Furthermore, Memory mechanisms and operations of various chargeable dielectrics and semiconductors for synaptic functions are also introduced. Through the designs of materials, device structures, and circuits, polymer-based artificial synapses will significantly impact the development of energy-efficient neuromorphic computing.