  The first goal of this paper is to construct examples of higher dimensional
contact manifolds with specific properties. Our main results in this direction
are the existence of tight virtually overtwisted closed contact manifolds in
all dimensions and the fact that every closed contact 3-manifold, which is not
(smoothly) a rational homology sphere, contact--embeds with trivial normal
bundle inside a hypertight closed contact 5-manifold. This uses known
construction procedures by Bourgeois (on products with tori) and Geiges (on
branched covering spaces). We pass from these procedures to definitions; this
allows to prove a uniqueness statement in the case of contact branched
coverings, and to study the global properties (such as tightness and
fillability) of the results of both constructions without relying on any
auxiliary choice in the procedures. A second goal allowed by these definitions
is to study relations between these constructions and the notions of supporting
open book, as introduced by Giroux, and of contact fiber bundle, as introduced
by Lerman. For instance, we give a definition of Bourgeois contact structures
on flat contact fiber bundles which is local, (strictly) includes the results
of the Bourgeois construction, and allows to recover an isotopy class of
supporting open books on the fibers. This last point relies on a
reinterpretation, inspired by an idea by Giroux, of supporting open books in
terms of pairs of contact vector fields.
