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Electroencephalography (EEG) produces time-series data of neural oscillations in the brain, and is one of the most commonly used methods for investigating both normal brain functions and brain disorders. Quantitative EEG analysis enables identification of frequencies and brain activity that are activated or impaired. With studies on the structural and functional networks of the brain, the concept of the brain as a complex network has been fundamental to understand normal brain functions and the pathophysiology of various neurological disorders. Functional connectivity is a measure of neural synchrony in the brain network that refers to the statistical interdependency between neural oscillations over time. In this review, we first discuss the basic methods of EEG analysis, including preprocessing, spectral analysis, and functional-connectivity and graph-theory measures. We then review previous EEG studies of brain network characterization in several neurological disorders, including epilepsy, Alzheimer’s disease, dementia with Lewy bodies, and idiopathic rapid eye movement sleep behavior disorder. Identifying the EEG-based network characteristics might improve the understanding of disease processes and aid the development of novel therapeutic approaches for various neurological disorders.