  We review the current status of heavy quarkonium production phenomenology
based on nonrelativistic effective field theories, focusing on spin-triplet
$S$-wave states such as $J/\psi$, $\psi(2S)$, and $\Upsilon$. We present some
representative examples for heavy quarkonium production mechanisms proposed in
the literature, which vary significantly depending on the choice of data
employed in analyses. We then discuss the r\^{o}le of polarization in
discriminating between the different possible scenarios for quarkonium
production. Other observables that may be useful in pinpointing the production
mechanism are also introduced, such as the $\eta_c$ production, associated
production of $J/\psi$ plus a gauge boson, and $J/\psi$ production at the
Electron-Ion Collider.
