Molecular communication is a novel approach for data transmission between miniaturized devices, especially in contexts where electrical signals are to be avoided. The communication is based on sending molecules (or other particles) at nano scale through channel instead sending electrons over a wire. Molecular communication devices have a large potential in medical applications as they offer an alternative to antenna-based transmission systems that may not be applicable due to size, temperature, or radiation constraints. The communication is achieved by transforming a digital signal into concentrations of molecules. These molecules are then detected at the other end of the communication channel and transformed back into a digital signal. Accurately modeling the transmission channel is often not possible which may be due to a lack of data or time-varying parameters of the channel (e. g., the movements of a person wearing a medical device). This makes demodulation of the signal very difficult. Many approaches for demodulation have been discussed with one particular approach having tremendous success: artificial neural networks. These networks imitate the decision process in the human brain and are capable of reliably classifying noisy input data. Training such a network relies on a large set of training data. As molecular communication as a technology is still in its early development phase, this data is not always readily available. We discuss neural network-based demodulation approaches relying on synthetic data based on theoretical channel models as well as works using actual measurements produced by a prototype test bed. In this work, we give a general overview over the field molecular communication, discuss the challenges in the demodulations process of transmitted signals, and present approaches to these challenges that are based on artificial neural networks.