This article explores the Kuramoto model describing the synchronization of a population of coupled oscillators. Two versions of this model are considered: a discrete version suitable for a population with a finite number of oscillators, and a continuum model found in the limit of an infinite population. When the strength of the coupling between the oscillators exceeds a threshold, the oscillators partially synchronize. We explore the transition in the continuum model, which takes the form of a bifurcation of a discrete mode from a continuous spectrum. We use numerical methods and perturbation theory to study the patterns of synchronization that form beyond transition, and compare with the synchronization predicted by the discrete model. There are similarities with instabilities in ideal plasmas and inviscid fluids, but these are superficial.