  We provide a set of tools for analyzing the geometry of elliptically fibered
Calabi-Yau manifolds, starting with a description of the total space rather
than with a Weierstrass model or a specified type of fiber/base. Such an
approach to the subject of F-theory compactification makes certain geometric
properties, which are usually hidden, manifest. Specifically, we review how to
isolate genus-one fibrations in such geometries and then describe how to find
their sections explicitly. This includes a full parameterization of the
Mordell-Weil group where non-trivial. We then describe how to analyze the
associated Weierstrass models, Jacobians and resolved geometries. We illustrate
our discussion with concrete examples which are complete intersections in
products of projective spaces (CICYs). The examples presented include cases
exhibiting non-abelian symmetries and higher rank Mordell-Weil group. We also
make some comments on non-flat fibrations in this context. In a companion paper
[1] to this one, these results will be used to analyze the consequences for
string dualities of the ubiquity of multiple fibrations in known constructions
of Calabi-Yau manifolds.
