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Speech processing technologies are omnipresent in our daily communication products and services. Neural networks, as powerful data-driven models, have shown promising performance in various research fields, including speech processing. This thesis focuses on neural network-based speech processing, and it can be divided into three parts as follows.In the field of speech prediction, a nonlinear speech predictor using the echo state network (ESN) is proposed as a novel adaptive prediction approach. This proposed nonlinear predictor shows better prediction performance than all baseline prediction methods in the simulations, including a predictor based on a long short-term memory (LSTM) structure. Second, the field of neural network-based speech enhancement puts focus on loss functions. A novel perceptual weighting filter (PWF) loss function motivated by the weighting filter from code-excited linear prediction (CELP) speech coding is proposed. A fully connected neural network (FCNN) and a convolutional neural network (CNN) are both used to evaluate the proposed loss functions, and the simulation results show their superior performance compared to baselines. Finally, neural network-based postprocessing for the enhancement of coded speech is studied. CNN-based postprocessors are proposed either to directly enhance the raw waveform in an end-to-end fashion, or to enhance the cepstral domain features using analysis synthesis. Furthermore, an advanced network structure, the fully convolutional recurrent network (FCRN), is utilized to enhance coded speech in the frequency domain, with the PWF loss function advantageously applied. The experimental results confirm the effectiveness of the proposed postprocessors with improved speech quality.