  We develop the properties of Weyl geometry, beginning with a review of the
conformal properties of Riemannian spacetimes. Decomposition of the Riemann
curvature into trace and traceless parts allows an easy proof that the Weyl
curvature tensor is the conformally invariant part of the Riemann curvature,
and shows the explicit change in the Ricci and Schouten tensors required to
insure conformal invariance. We include a proof of the well-known condition for
the existence of a conformal transformation to a Ricci-flat spacetime. We
generalize this to a derivation of the condition for the existence of a
conformal transformation to a spacetime satisfying the Einstein equation with
matter sources. Then, enlarging the symmetry from Poincar\'e to Weyl, we
develop the Cartan structure equations of Weyl geometry, the form of the
curvature tensor and its relationship to the Riemann curvature of the
corresponding Riemannian geometry. We present a simple theory of Weyl-covariant
gravity based on a curvature-linear action, and show that it is conformally
equivalent to general relativity. This theory is invariant under dilatations,
but not the full conformal group.
