The theoretical reasoning that leads to the prediction of, among others, total cross section rising like (ℓn s)2is reviewed. Briefly, relativistic kinematics dictate the dependence on longitudinal momentum of particles produced in the pionization region, and leads to a factor of ℓn s in the integrated cross section. Unitarity in the t‐channel then amplifies this ℓn s to be a power of s, divided possibly by a power of ℓn s. This power of s is a realization of the strongly absorptive potential with a coupling constant increasing with energy, and thus the Froissart bound for total cross sections is saturated. Many other results, concerning in particular the phases of the scattering amplitudes in the forward direction and differential cross sections for both elastic and inelastic diffractive scattering, are obtained at the same time.In order to make quantitative comparison with experimental data, past and future, we describe a simple model of hadron scattering. When applied to the elastic scattering of pp,pp, &pgr;±p, and K±p, this model describes reasonably well existing data beyond 20GeV. From the resulting parameters, quantitative predictions about these processes are given in detail up to the highest energy that may become available from accelerators in this century, namely 1.5TeV on 1.5TeV. These considerations are also extended to inelastic diffractive scattering without spin‐parity exchange.A number of problems related to increasing total cross sections, some unsolved, are discussed.