  We argue that conformal invariance in flat spacetime implies Weyl invariance
in a general curved background metric for all unitary theories in spacetime
dimensions $d \leq 10$. We also study possible curvature corrections to the
Weyl transformations of operators, and show that these are absent for operators
of sufficiently low dimensionality and spin. We find possible `anomalous' Weyl
transformations proportional to the Weyl (Cotton) tensor for $d > 3$ ($d = 3$).
The arguments are based on algebraic consistency conditions similar to the
Wess-Zumino consistency conditions that classify possible local anomalies. The
arguments can be straightforwardly extended to larger operator dimensions and
higher $d$ with additional algebraic complexity.
