Electroencephalogram (EEG) signals acquired from brain can provide an effective representation of the human's physiological and pathological states. Up to now, much work has been conducted to study and analyze the EEG signals, aiming at spying the current states or the evolution characteristics of the complex brain system. Considering the complex interactions between different structural and functional brain regions, brain network has received a lot of attention and has made great progress in brain mechanism research. In addition, characterized by autonomous, multi-layer and diversified feature extraction, deep learning has provided an effective and feasible solution for solving complex classification problems in many fields, including brain state research. Both of them show strong ability in EEG signal analysis, but the combination of these two theories to solve the difficult classification problems based on EEG signals is still in its infancy. We here review the application of these two theories in EEG signal research, mainly involving brain-computer interface, neurological disorders and cognitive analysis. Furthermore, we also develop a framework combining recurrence plots and convolutional neural network to achieve fatigue driving recognition. The results demonstrate that complex networks and deep learning can effectively implement functional complementarity for better feature extraction and classification, especially in EEG signal analysis.