We give an overview of the present status of knowledge of the production of J/ψ, ψ ′ and Υ 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 experimental 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/ψ, ψ ′ and Υ 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/ψ, ψ ′ and Υ production at nonzero p T .(where the sum stands for S, L, J and the colour. The long-distance matrix element (LDME) O Q [ 2S+1 L (1,8) J ] takes account of the transition between the QQ pair and the final physical state Q. Its power scaling rule comes both from the suppression in v of the Fock-state component [ 2S+1 L (1,8) J ] in the wave function of Q and from the scaling of the NRQCD interaction responsible for the transition. Usually, one defines O Q [ 2S+1 L (1,8) J (1,8) J ] to the dominant quarkonium Q Fock state. e The second line of Eq. (26) is nothing but a short way of expressing this operator. J/ψ, ψ ′ and Υ Production at Hadron Colliders: a review 33The idea of combining fragmentation as the main source of production with allowed transitions between a χ c to a 3 S 1 in a colour-octet state, was applied by Braaten and Yuan 45 . Indeed, similar formulae to the one written for fragmentation within the CSM can be written for fragmentation functions in NRQCD 78 :