Abstract: In this talk, I will confront the problem of the enhancement of conformal invariance in flat spacetime to Weyl invariance in curved spacetime, extending the equivalence of the conformal and Weyl symmetries to d ≤ 10 spacetime dimensions. I will put forward a compelling argument for the statement that for all unitary theories in d ≤ 10, conformal invariance in flat spacetime implies Weyl invariance in a general curved background metric. In addition, I will examine possible curvature corrections to the Weyl transformation laws of operators and show that these are in fact absent for operators of sufficiently low dimensionality and spin. In particular, I will demonstrate this for an important class of operators, namely relevant scalar operators in d ≤ 6, and establish that the Weyl transformations of these operators are the canonical ones. Further, I will identify a class of consistent ‘anomalous’ curvature corrections proportional to the Weyl (Cotton) tensor in d > 3 (d = 3). The arguments rely on algebraic consistency conditions reminiscent of the famous Wess-Zumino consistency conditions employed for the classification of Weyl anomalies. They can be extended to higher d and for more general operators at the price of greater algebraic complexity.