We present some preliminary results on the next-to-leading order calculation in QCD of quarkonium production cross sections in hadronic collisions. We will show that the NLO total cross sections for P-wave states produced at high energy are not reliable, due to the appearance of very large and negative contributions. We also discuss some issues related to the structure of final states in colour-octet production and to high-pt fragmentation.
Quarkonium production in pNRQCD: the P-wave case