  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 SU(1,1)/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 non-conformal theories.
