We have investigated several models of Pomeron and Odderon contributions to high energy elastic $pp$ and $\bar p p$ scattering. The questions we address concern their role in this field, the behavior of the scattering amplitude (or of the total cross-section) at high energy, and how to fit all high energy elastic data. The data are extremely well reproduced by our approach at all momenta and for sufficiently high energies. The relative virtues of Born amplitudes and of different kinds of eikonalizations are considered. An important point in this respect is that secondary structures are predicted in the differential cross-sections at increasing energies and these phenomena appear quite directly related to the procedure of eikonalizing the various Born amplitudes. We conclude that these secondary structures arise naturally within the eikonalized procedure (although their precise localization turns out to be model dependent). The fitting procedure naturally predicts the appearance of a zero at small $|t|$ in the real part of the even amplitude as anticipated by general theorems. We would like to stress, once again, how important it would be to have at LHC both $pp$ and $p \bar p$ options for many questions connected to the general properties of high energy hadronic physics and for a check of our predictions.

28 pages, LaTeX, 7 figures