  We develop a general formalism of duality rotations for bosonic conformal
spin-$s$ gauge fields, with $s\geq 2$, in a conformally flat four-dimensional
spacetime. In the $s=1$ case this formalism is equivalent to the theory of
$\mathsf{U}(1)$ duality-invariant nonlinear electrodynamics developed by
Gaillard and Zumino, Gibbons and Rasheed, and generalised by Ivanov and Zupnik.
For each integer spin $s\geq 2$ we demonstrate the existence of families of
conformal $\mathsf{U}(1)$ duality-invariant models, including a generalisation
of the so called ModMax Electrodynamics ($s=1$). Our bosonic results are then
extended to the $\mathcal{N}=1$ and $\mathcal{N}=2$ supersymmetric cases. We
also sketch a formalism of duality rotations for conformal gauge fields of
Lorentz type $(m/2, n/2)$, for positive integers $m $ and $n$.
