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In this work, we study F-theory compactifications on manifolds. Such geometries, which we study in both 4- and 6dimensions, are both ubiquitous within the set of Calabi-Yau manifolds and play a crucial role in heterotic/F-theory duality. We discuss the most general formulation of P1-bundles of this type, as well as fibrations which degenerate at higher codimension loci. In the course of this study, we find a number of new phenomena. For example, in both 4- and 6-dimensions we find transitions whereby the base of a P1-bundle can change nature, or "jump ", at certain loci in complex structure moduli space. We discuss the implications of this jumping for the associated heterotic duals. We argue that P1-bundles with only rational sections lead to heterotic duals where the Calabi-Yau manifold is elliptically fibered over the section of the P1 bundle, and not its base. As expected, we see that degenerations of the P1 fibration of the F-theory base correspond to 5-branes in the dual heterotic physics, with the exception of cases in which the fiber degenerations exhibit monodromy. Along the way, we discuss a set of useful formulae and tools for describing F-theory compactifications on this class of Calabi-Yau manifolds.