The Kuramoto model is a paradigmatic model for studying synchronization phenomena in large populations of interacting elements. This review covers the mathematical treatment, numerical methods, and various extensions of the original model. The model describes a population of phase oscillators with natural frequencies distributed according to a probability density function \( g(\omega) \). The dynamics are governed by the Kuramoto equation, which includes coupling terms that depend on the phase differences between oscillators. Key aspects of the Kuramoto model include: 1. **Stationary Synchronization**: For mean-field coupling, the model exhibits stationary synchronization when the coupling strength \( K \) exceeds a critical value \( K_c \). The order parameter \( r \) measures the coherence of the oscillator population, with \( r = 0 \) corresponding to incoherence and \( r = 1 \) to global synchronization. 2. **Stability of Solutions**: The stability of incoherent and partially synchronized states is analyzed using linear stability theory. Incoherence is neutrally stable for \( K < K_c \) and linearly unstable for \( K > K_c \). The stability of partially synchronized states is less well understood. 3. **Finite-Size Effects**: The behavior of a finite number of oscillators is different from the limit of infinitely many oscillators. Finite-size effects can lead to unexpected phenomena, such as time-periodic synchronization. 4. **Noisy Kuramoto Model**: Adding white-noise forcing terms to the mean-field Kuramoto model results in a nonlinear Fokker-Planck equation. This model is more physically realistic but mathematically more challenging to analyze. 5. **Phase Diagram**: The phase diagram of the Kuramoto model depends on the natural frequency distribution \( g(\omega) \). For unimodal distributions, the stability of incoherence and synchronization can be determined by the coupling strength \( K \) and diffusion constant \( D \). 6. **Extensions and Applications**: The Kuramoto model has been extended to include short-range couplings, disorder, time delays, external fields, and multiplicative noise. It has found applications in various fields, including neural networks, Josephson junctions, laser arrays, and chemical oscillators. The review also discusses numerical methods for solving the Kuramoto model, including finite-difference and spectral methods, and the moments approach. It concludes with a discussion of open problems and future research directions.The Kuramoto model is a paradigmatic model for studying synchronization phenomena in large populations of interacting elements. This review covers the mathematical treatment, numerical methods, and various extensions of the original model. The model describes a population of phase oscillators with natural frequencies distributed according to a probability density function \( g(\omega) \). The dynamics are governed by the Kuramoto equation, which includes coupling terms that depend on the phase differences between oscillators. Key aspects of the Kuramoto model include: 1. **Stationary Synchronization**: For mean-field coupling, the model exhibits stationary synchronization when the coupling strength \( K \) exceeds a critical value \( K_c \). The order parameter \( r \) measures the coherence of the oscillator population, with \( r = 0 \) corresponding to incoherence and \( r = 1 \) to global synchronization. 2. **Stability of Solutions**: The stability of incoherent and partially synchronized states is analyzed using linear stability theory. Incoherence is neutrally stable for \( K < K_c \) and linearly unstable for \( K > K_c \). The stability of partially synchronized states is less well understood. 3. **Finite-Size Effects**: The behavior of a finite number of oscillators is different from the limit of infinitely many oscillators. Finite-size effects can lead to unexpected phenomena, such as time-periodic synchronization. 4. **Noisy Kuramoto Model**: Adding white-noise forcing terms to the mean-field Kuramoto model results in a nonlinear Fokker-Planck equation. This model is more physically realistic but mathematically more challenging to analyze. 5. **Phase Diagram**: The phase diagram of the Kuramoto model depends on the natural frequency distribution \( g(\omega) \). For unimodal distributions, the stability of incoherence and synchronization can be determined by the coupling strength \( K \) and diffusion constant \( D \). 6. **Extensions and Applications**: The Kuramoto model has been extended to include short-range couplings, disorder, time delays, external fields, and multiplicative noise. It has found applications in various fields, including neural networks, Josephson junctions, laser arrays, and chemical oscillators. The review also discusses numerical methods for solving the Kuramoto model, including finite-difference and spectral methods, and the moments approach. It concludes with a discussion of open problems and future research directions.