In this thesis we gave a basic theoretical introduction to neural networks and motivated their practical use. After that, we precisely worked out an alogorithm for training these networks on data. Next, we analyzed recurrent neural networks in more detail, worked out the main problems with their training and introduced more advanced versions of these networks which are the LSTM and GRU networks. Afterwards, we introduced the problem of music source separation and described it mathematically. We introduced basic results about Fourier transformations which allowed us to model this problem more easily. We also derived an algorithm for faster calculation of the discrete Fourier transform which made practical use of this transformation easier. These results led us to the definition of the spectrogram as an appropriate way of describing audio data with the goal of training neural networks on it. Finally, we defined a neural network architecture based on recurrent networks and applied it to the problem of music source separation. At the very end, we described the results of this model and briefly analyzed them.