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The entrainment transition of coupled random frequency oscillators presents along-standing problem in nonlinear physics. The onset of entrainment inpopulations of large but finite size exhibits strong sensitivity tofluctuations in the oscillator density at the synchronizing frequency. This isthe source for the unusual values assumed by the correlation size exponent$\nu'$. Locally coupled oscillators on a $d$-dimensional lattice exhibit twotypes of frequency entrainment: symmetry-breaking at $d > 4$, and aggregationof compact synchronized domains in three and four dimensions. Various criticalproperties of the transition are well captured by finite-size scaling relationswith simple yet unconventional exponent values.

To synchronize or not to synchronize, that is the question: finite-size scaling and fluctuation effects in the Kuramoto model