  We give an overview of the present status of knowledge of the production of
J/psi, psi' and Upsilon in high-energy hadron collisions. We first present two
early models, namely the Colour-Singlet Model (CSM) and the Colour-Evaporation
Model (CEM). The first is the natural application of pQCD to quarkonium
production and has been shown to fail dramatically to describe data, the second
is its phenomenological counterpart and was introduced in the spirit of the
quark-hadron duality in the late seventies. Then, we expose the most recent
experimental measurements of J/psi, psi' and Upsilon prompt and direct
production at nonzero p_T from two high-energy hadron colliders, the Tevatron
and RHIC. In a third part, we review six contemporary models describing J/psi,
psi' and Upsilon production at nonzero p_T.

High Energy Physics - Phenomenology;High Energy Physics - Experiment;Nuclear Theory