  We give predictions of $\jpsi$ and $\psits$ yields and polarizations in
prompt production at hadron colliders based on non-relativistic QCD
factorization formula. We calculate short-distance coefficients of all
important color-octet intermediate channels as well as color-singlet channels
up to $\mathcal{O}(\alpha_S^4)$, i.e. next-to-leading order in $\alpha_S$. For
prompt $\jpsi$ production, we also take into account feeddown contributions
from $\chi_{cJ}$(J=0,1,2) and $\psits$ decays. Color-singlet long-distance
matrix elements (LDMEs) are estimated by using potential model, and color-octet
LDMEs are extracted by fitting the Tevatron yield data only. The predictions
are satisfactory for both yields and polarizations of prompt $\jpsi$ and prompt
$\psits$ production at the Tevatron and the LHC. In particular, we find our
predictions for polarizations of prompt $\jpsi$ production have only a little
difference from our previous predictions for polarizations of direct $\jpsi$
production.
