  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-p_T fragmentation.
