<p>This review considers state-of-the-art methods for analyzing functional brain networks recovered from functional magnetic resonance imaging (fMRI) data. The focus is on the application of mathematical and computational approaches, including complex network theory and machine learning methods, to study topological properties and differences of functional connectivity in the brain. Various functional network recovery methods such as correlation, information, regression, and pattern-based approaches and their application to the diagnosis of neurological and psychiatric disorders, including Alzheimer’s disease, schizophrenia, and depression, are discussed. Special attention is paid to the problem of intersubject variability of data and methods to overcome it, including the concept of consensus networks. Prospects for using machine learning, in particular graph neural networks, to classify functional networks and identify disease biomarkers, are considered. Finally, future research directions, including the study of higher-order interactions in functional brain networks, which may lead to the development of new diagnostic and therapeutic strategies, are discussed.</p>