Construction of all $\mathcal{N}=4$ conformal supergravities

All $N=4$ conformal supergravities in four space-time dimensions are constructed. These are the only $N=4$ supergravity theories whose actions are invariant under off-shell supersymmetry. They are encoded in terms of a holomorphic function that is homogeneous of zeroth degree in scalar fields that parametrize an $\mathrm{SU}(1,1)/\mathrm{U}(1)$ coset space. When this function equals a constant the Lagrangian is invariant under continuous SU(1,1) transformations. The construction of these higher-derivative invariants also opens the door to various applications for nonconformal theories.