Aspects of the Fundamental Theory of Proton-Proton Scattering

After recalling the existence of a high energy bound on proton-proton total cross-sections, we discuss the various phenomena which occur when these cross-sections rise and especially when they have the qualitative behaviour of the bound: rising elastic cross-sections, shrinking diffraction peak, validity of the Pomeranchuk theorem for total and elastic cross-sections, existence of a positive real part of the forward amplitude at high energies.