  The equations of General Relativity are recast in the form of a wave equation
for the Weyl tensor. This allows to reformulate gravitational wave theory in
terms of curvature waves, rather than metric waves. The existence of two
transverse polarization states for curvature waves is proven and in the
linearized approximation the quadrupole formula is rederived. A perturbative
scheme to extend the linearized result to the non-linear regime is outlined.
