  After an introduction motivating the study of quarkonium production, we
review the recent developments in the phenomenology of quarkonium production in
inclusive scatterings of hadrons and leptons. We naturally address data and
predictions relevant for the LHC, the Tevatron, RHIC, HERA, LEP, B factories
and EIC. An up-to-date discussion of the contributions from feed downs within
the charmonium and bottomonium families as well as from b hadrons to charmonia
is also provided. This contextualises an exhaustive overview of new observables
such as the associated production along with a Standard Model boson (photon, W
and Z), with another quarkonium, with another heavy quark as well as with light
hadrons or jets. We address the relevance of these reactions in order to
improve our understanding of the mechanisms underlying quarkonium production as
well as the physics of multi-parton interactions, in particular the double
parton scatterings. An outlook towards future studies and facilities concludes
this review.
