Towards the saturation of the Froissart bound

It is the aim of this paper to summarize results about the construction of amplitudes, which rigorously satisfy Mandelstam analyticity, crossing symmetry, and (at least partly) the constraints imposed by elastic and inelastic unitarity. The results are discussed under particular emphasis of a strong increase of the absorptive part of the forward amplitude and the saturation of the Froissart bound.