  We introduce an appropriate formalism in order to study conformal Killing
(symmetric) tensors on Riemannian manifolds. We reprove in a simple way some
known results in the field and obtain several new results, like the
classification of conformal Killing $2$-tensors on Riemannian products of
compact manifolds, Weitzenb\"ock formulas leading to non-existence results, and
construct various examples of manifolds with conformal Killing tensors.
