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 and nuclear collisions 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.