Deep neural networks applied to signal processing problems will have to incorporate various architectural features to remember the history of the input signals, e.g., loops between the layers, “gated” neurons, and tapped delay lines. But real brains have rich dynamics expressed in terms of various frequency bands (alpha, beta, gamma, delta), exhibiting dynamical phenomena like phase locking, synchronization, etc. A typical Recurrent Neural Network or RNN-type model has serious shortcomings in representing these dynamic aspects of the brain. In this paper, we propose a novel class of bio-inspired deep neural network models known as deep oscillatory neural networks. There are deep networks of nonlinear oscillators trained on lines similar to a deep network. We present two variations of the models − the Deep Oscillatory Neural Network (DONN) and a convolutional variation of it named Oscillatory Convolutional Neural Network (OCNN) – and apply the models to a variety of problems involving classification and prediction of Electroencephalogram (EEG) signals: prediction of nearby EEG channels, classification of single-channel EEG data, and classification of spatiotemporal topographical images of EEG signal. Simulations show that the proposed oscillatory neural networks yield superior or comparable classification and prediction accuracy levels compared to published models, while requiring substantially smaller number of trainable parameters. The potential for creating energy-efficient hardware realizations of oscillator networks suggests an added incentive to intensify study of oscillator networks for classification problems.