**ABSTRACT:**

Let A be a Weyl operator with symbol a: A <-Weyl-> a. It is

well-known that Ŝ^(-1) AŜ <-Weyl-> a o S for every linear symplectic automorphism

S where Ŝ is anyone of the two elements of the metaplectic group covering

S. We ask the question whether this covariance property can be extended to

larger classes of pseudo-differential operators or to larger groups of automor-

phisms (symplectic, or not, linear, or not) of phase space (R^2d; sigma). We show

that the answer is negative in both cases: symplectic covariance of Weyl op-

erators cannot be extended to the non-linear case, moreover the symplectic

group is the largest group of linear operators having this property. More-

over this property characterizes the Weyl calculus, and cannot thus be fully

extended to other classes of pseudo-differntial operators. We brifly discuss

partial symplectic covariance for Shubin and Bornâ€“Jordan operators.