  The framework to compute the cross sections for the production of particles
with high mass and/or large transverse momentum in double- (DPS), triple-
(TPS), and in general $n$-parton scatterings, from the corresponding
single-parton ($\sigma_{\rm SPS}$) values in high-energy proton-proton,
proton-nucleus, and nucleus-nucleus is reviewed. The basic parameter of the
factorized $n$-parton scattering ansatz is an effective cross section
$\sigma_{\rm eff}$ encoding all unknowns about the underlying generalized
$n$-parton distribution in the proton (nucleon). In its simplest and most
economical form, the $\sigma_{\rm eff}$ parameter can be derived from the
transverse parton profile of the colliding protons and/or nucleus, using a
Glauber approach. Numerical examples for the cross sections and yields expected
for the concurrent DPS or TPS production of heavy-quarks, quarkonia, and/or
gauge bosons in proton and nuclear collisions at LHC and Future Circular
Collider (FCC) energies are provided. The obtained cross sections are based on
perturbative QCD predictions for $\sigma_{\rm SPS}$ at next-to-leading-order
(NLO) or next-to-NLO (NNLO) accuracy including, when needed, nuclear
modifications of the corresponding parton densities.
