  We revisit Weyl geometry in the context of recent higher-dimensional theories
of spacetime. After introducing the Weyl theory in a modern geometrical
language we present some results that represent extensions of Riemannian
theorems. We consider the theory of local embeddings and submanifolds in the
context of Weyl geometries and show how a Riemannian spacetime may be locally
and isometrically embedded in a Weyl bulk. We discuss the problem of classical
confinement and the stability of motion of particles and photons in the
neighbourhood of branes for the case when the Weyl bulk has the geometry of a
warped product space. We show how the confinement and stability properties of
geodesics near the brane may be affected by the Weyl field. We construct a
classical analogue of quantum confinement inspired in theoretical-field models
by considering a Weyl scalar field which depends only on the extra coordinate.
