Do we understand near-forward elastic scattering up to TeV energies?

In 1970, on purely theoretical grounds, all total hadronic total cross sections were predicted to increase without limit for higher and higher energies. This was contrary to the conventional belief at that time. In 1978, an accurate phenomenological model was formulated for the case of proton-proton and antiproton-proton interactions. The parameters for this model were slightly improved in 1984 using the additional available experimental data. Since then, for thirty years these parameters have not changed. This development, including especially the difficult task of formulating this phenomenological model and the comparison of the predictions of this model with later experimental results, is summarized.