We revisit Weyl geometry in the context of recent higher-dimensional theories of space-time. 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 space-time 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 neighborhood 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 analog of quantum confinement inspired in theoretical-field models by considering a Weyl scalar field which depends only on the extra coordinate.