One of the challenging problem in heavy flavor physics is the production of heavy quarkonia at hadron colliders. The J/psi and psi' production cross sections at high p_T observed by CDF at the Tevatron are more than one order of magnitude larger than conventional QCD predictions. In the framework of nonrelativistic QCD (NRQCD), the color-octet gluon fragmentation might increase the cross sections, but the predicted trasverse polarizations of J/psi and psi' are in contradiction with the measurements. We show that the next-to-leading order (NLO) QCD corrections including both the color-singlet and color-octet channels may essentially change the large p_T behavior of J/psi, and that the predicted cross sections can be in good agreement with the Tevatron data and the newly observed LHC data at large p_T. Moreover, the J/psi polarization puzzle may also be explained. However, we find that the long-distance matrix elements (LDMEs) extracted from hadron colliders are not compatible with that extracted from the e+e- annihilation data observed by Bell at the B factory. Furhther theoretical investigation and experimental measurement are needed to clarify the problem. Puzzles in the P-wave charmonium states chi_cJ are also discussed.