Natural organic materials such as protein and carbohydrates are abundant in nature, renewable, and biodegradable, desirable for the construction of artificial synaptic devices for emerging neuromorphic computing systems with energy efficient operation and environmentally friendly disposal. These artificial synaptic devices are based on memristors or transistors with the memristive layer or gate dielectric formed by natural organic materials. The fundamental requirement for these synaptic devices is the ability to mimic the memory and learning behaviors of biological synapses. This paper reviews the synaptic functions emulated by a variety of artificial synaptic devices based on natural organic materials and provides a useful guidance for testing and investigating more of such devices.