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In this review, we put forward a common framework in which we discuss in a unified method known outcomes with more modern developments obtained for a generalized Kuramoto mannequin that features inertial effects and noise. We describe the model from a different perspective, highlighting the long-range nature of the interplay between the oscillators, and emphasizing the equilibrium and out-of-equilibrium features of its dynamics from a statistical physics perspective. In this evaluation, we first introduce the mannequin and focus on both for the noiseless and noisy dynamics and for unimodal frequency distributions the synchronization transition that occurs within the stationary state. We then introduce …