This review discusses the application of graph theory to neuroscience, focusing on the analysis of complex networks in the brain. Since the discovery of small-world and scale-free networks, the study of complex systems from a network perspective has advanced significantly. The relationship between the structural properties of networks and the dynamics on these networks has been a key area of research. For example, the 'synchronizability' of complex networks of coupled oscillators can be determined by graph spectral analysis. These developments have inspired new applications in neuroscience, including the study of neural networks, anatomical connectivity, and functional connectivity based on fMRI, EEG, and MEG. These studies suggest that the human brain can be modeled as a complex network with a small-world structure at both the anatomical and functional levels. This small-world structure is hypothesized to reflect an optimal situation associated with rapid synchronization and information transfer, minimal wiring costs, and a balance between local processing and global integration. The topological structure of functional networks is likely influenced by genetic and anatomical factors but can be modified during tasks. There is also increasing evidence that various brain diseases, such as Alzheimer's disease, schizophrenia, brain tumors, and epilepsy, may be associated with deviations from the optimal small-world pattern in functional network topology. The human brain is considered the most complex object in the universe. Understanding its intricate wiring patterns and how they give rise to normal and disturbed brain function is one of the most challenging areas in modern science. In the last decades of the 20th century, significant progress was made in neuroscience with a reductionistic, molecular biological research program. However, despite the impressive increase in knowledge in terms of molecular and genetic mechanisms, progress in true understanding has been disappointing, and few theories are available that attempt to explain higher-level brain processes. There has been increased interest in searching for other approaches to study brain processes and their relation to consciousness and higher brain functions. One strategy has been to conceive the brain as a complex dynamical system and to search for new approaches in other fields of science that are also devoted to the study of complex systems. In recent years, considerable progress has been made in the study of general complex systems, consisting of large numbers of weakly interacting elements. Three research areas in physics and mathematics have proven to be particularly valuable in the study of complex systems: (i) nonlinear dynamics and related areas such as synergetics; (ii) statistical physics which deals with universal phenomena at phase transitions and scaling behaviour; and (iii) the modern theory of networks, which is derived from graph theory. Nonlinear dynamics has been applied to the study of the brain since 1985, and has become a very active research field in itself. Application of nonlinear dynamics to neuroscience has led to the introduction of new concepts such as attractors, control parameters, and bifurcations as well as to the development of a whole range of new analytical tools to extract nonlinear properties from time series of brain activity.This review discusses the application of graph theory to neuroscience, focusing on the analysis of complex networks in the brain. Since the discovery of small-world and scale-free networks, the study of complex systems from a network perspective has advanced significantly. The relationship between the structural properties of networks and the dynamics on these networks has been a key area of research. For example, the 'synchronizability' of complex networks of coupled oscillators can be determined by graph spectral analysis. These developments have inspired new applications in neuroscience, including the study of neural networks, anatomical connectivity, and functional connectivity based on fMRI, EEG, and MEG. These studies suggest that the human brain can be modeled as a complex network with a small-world structure at both the anatomical and functional levels. This small-world structure is hypothesized to reflect an optimal situation associated with rapid synchronization and information transfer, minimal wiring costs, and a balance between local processing and global integration. The topological structure of functional networks is likely influenced by genetic and anatomical factors but can be modified during tasks. There is also increasing evidence that various brain diseases, such as Alzheimer's disease, schizophrenia, brain tumors, and epilepsy, may be associated with deviations from the optimal small-world pattern in functional network topology. The human brain is considered the most complex object in the universe. Understanding its intricate wiring patterns and how they give rise to normal and disturbed brain function is one of the most challenging areas in modern science. In the last decades of the 20th century, significant progress was made in neuroscience with a reductionistic, molecular biological research program. However, despite the impressive increase in knowledge in terms of molecular and genetic mechanisms, progress in true understanding has been disappointing, and few theories are available that attempt to explain higher-level brain processes. There has been increased interest in searching for other approaches to study brain processes and their relation to consciousness and higher brain functions. One strategy has been to conceive the brain as a complex dynamical system and to search for new approaches in other fields of science that are also devoted to the study of complex systems. In recent years, considerable progress has been made in the study of general complex systems, consisting of large numbers of weakly interacting elements. Three research areas in physics and mathematics have proven to be particularly valuable in the study of complex systems: (i) nonlinear dynamics and related areas such as synergetics; (ii) statistical physics which deals with universal phenomena at phase transitions and scaling behaviour; and (iii) the modern theory of networks, which is derived from graph theory. Nonlinear dynamics has been applied to the study of the brain since 1985, and has become a very active research field in itself. Application of nonlinear dynamics to neuroscience has led to the introduction of new concepts such as attractors, control parameters, and bifurcations as well as to the development of a whole range of new analytical tools to extract nonlinear properties from time series of brain activity.