Nowadays, neural networks are actively used for data hiding; however, there is currently no systematic knowledge regarding their utilization in this field. This is a significant gap, considering that neural network-based data hiding has already formed a large and quite independent area of research. This review aims to provide such systematization. It also provides a general framework of neural network usage for data hiding, comparisons of quantitative and qualitative indicators of the effectiveness of existing data hiding methods, and discussion and conclusions containing the most promising directions and tools for research in the field of data hiding using neural networks, as well as the main achievements and the persisting challenges. This work targets researchers looking for the most suitable data hiding method for their application, as well as the developers of data hiding methods.