  We develop a manifestly conformal approach to describe linearised
(super)conformal higher-spin gauge theories in arbitrary conformally flat
backgrounds in three and four spacetime dimensions. Closed-form expressions in
terms of gauge prepotentials are given for gauge-invariant higher-spin (super)
Cotton and (super) Weyl tensors in three and four dimensions, respectively. The
higher-spin (super) Weyl tensors are shown to be conformal primary
(super)fields in arbitrary conformal (super)gravity backgrounds, however they
are gauge invariant only if the background (super) Weyl tensor vanishes. The
proposed higher-spin actions are (super) Weyl-invariant on arbitrary curved
backgrounds, however the appropriate higher-spin gauge invariance holds only in
the conformally flat case. We also describe conformal models for generalised
gauge fields that are used to describe partially massless dynamics in three and
four dimensions. In particular, generalised higher-spin Cotton and Weyl tensors
are introduced.
