The B factories provide a unique playground for the study of the properties of conventional and exotic quarkonium states, We report on a variety of recent results obtained using the full data set collected with the BaBar detector at the PEPII e+e- collider. In particular, we present measurements of the prompt production of J/psi or psi(2S) in association with a second charmonium state at a center-of-mass energy of 10.58 GeV, searches for exotic neutral and charged charmonium-like states, studies of charmonium production in two-photon fusion and initial state radiation processes, as well as studies of radiative transitions between bottomonium states.