Neuromorphic devices that emulate biological neural systems have been actively studied to overcome the limitations of conventional von Neumann computing structure. Implementing various synaptic characteristics and decay time in the devices is important for various wearable neuromorphic applications. Polymer-based artificial synapses have been proposed as a solution to satisfy these requirements. Owing to the characteristics of polymer conjugated materials, such as easily tunable optical/electrical properties, mechanical flexibility, and biocompatibility, polymer-based synaptic devices are investigated to demonstrate their ultimate applications replicating biological nervous systems. In this review, we discuss various synaptic properties of artificial synaptic devices, including the operating mechanisms of synaptic devices. Furthermore, we review recent studies on polymer-based synaptic devices, focusing on strategies that modulate synaptic plasticity and synaptic decay time by changing the polymer structure and fabrication process. Finally, we show how the modulation of the synaptic properties can be applied to three major categories of these devices, including neuromorphic computing, artificial synaptic devices with sensing functions, and artificial nerves for neuroprostheses.