  Heavy quarkonium production in the framework of the nonrelativistic quantum
chromodynamics and leading order of the parton Reggeization approach at the
Tevatron and LHC Colliders is discussed. The new results, which are reviewed in
this report, include the comparison with recent data on chi_{c1,c2} production
from ATLAS Collaboration and special discussion of heavy quarkonium
polarization issues in the considered framework.
