Synchronizing systems -- Introduction -- The oscillators and their interaction: A qualitative discussion -- Oscillators as limit cycles -- Interacting limit-cycle oscillators -- Synchronizing systems as statistical mechanical systems -- The features of a statistical physical description -- Some results for noiseless interacting oscillators -- The oscillators with inertia -- Appendix 1: A two-dimensional dynamics with a limit-cycle attractor -- Appendix 2: The Lyapunov exponents -- Appendix 3: The one-body distribution function in an N-body system -- Oscillators with first-order dynamics -- The oscillators with distributed natural frequencies -- The Kuramoto model -- Unimodal symmetric g(w) -- Nonunimodal g(w) -- Appendix 1: An H-theorem for a particular simple case -- Appendix 2: Form of the function r(K) for symmetric and unimodal frequency distributions in the Kuramoto model -- Appendix 3: The numerical solution of Eq. (2.34) -- Oscillators with second-order dynamics -- Generalized Kuramoto model with inertia and noise -- Nonequilibrium first-order synchronization phase transition: Simulation results -- Analysis in the continuum limit: The Kramers equation -- Phase diagram: Comparison with numeric -- Appendix 1: The noiseless Kuramoto model with inertia: Connection with electrical power distribution models -- Appendix 2: Proof that the dynamics (3.9) does not satisfy detailed balance -- Appendix 3: Simulation details for the dynamics (3.9) -- Appendix 4: Derivation of the Kramers equation -- Appendix 5: Nature of solutions of Eq. (3.32) -- Appendix 6: Solution of the system of equations (3.39) -- Appendix 7: Convergence properties of the expansion (3.38).