Univalent logharmonic mappings (in the author's sense) are univalent solutions of the nonlinear elliptic partial differential equation \[ \overline{f_ z(z)}=a(z)(\overline{f(z)}/f(z))f_ z(z), \] where a(z) is an analytic function in the plane domain U and \(| a(z)| <1\) for all \(z\in U.\) This article is a survey of the authors' results. It contains a weaker form of the Riemann mapping theorem, conditions for the existence of logharmonic mappings of an arbitrary domain to canonical domains, its specialisation for starlike domains and some results about harmonic automorphisms.