Abstract: Two functions defined on a Riemannian manifold are said to be conjugate if at every point their gradients are mutually orthogonal and of the same length. In dimension 2, a function admits a conjugate if and only if it is harmonic. What happens in dimension 3? Is there a differential criterion for a function $f$ to admit a conjugate? The solution turns out to be quite tricky. This is joint work with Mike Eastwood.