Generalized eikonalization and unitarity

In the literature, the notion of eikonalization is often used as synonymous of unitarization or, at least, as implying that unitarity is not violated. This, to the very least, appears to be wishful thinking. We discuss the properties of various types of eikonalization within a unified treatment. Linear trajectories with intercept larger than unity (so popular nowadays) lead to small asymptotic violations of unitarity even after eikonalization. Classes of eikonalizations in which the Odderon could dominate over the Pomeron are given; even so the maximal Odderon is still excluded by eikonalization.