  We consider first order symmetry operators for the equations of motion of
differential $p$-form fields in general $D$-dimensional background geometry of
any signature for both massless and massive cases. For $p=1$ and $p=2$ we give
the general forms of the symmetry operators. Then we find a class of symmetry
operators for arbitrary $p$ and $D$, which is naturally suggested by the lower
$p$ results.
