It follows from the famous Weierstrass representation, for MINIMAL surfaces in $3$-dimensional Euclidean space, that any such surface can locally be presented by a (weakly) CONFORMAL HARMONIC immersion from the complex plane given in terms of two holomorphic functions. The inverse image of a regular value of a complex-valued holomorphic function on a Kähler manifold is a MINIMAL submanifold of codimension $2$. It is a direct consequence of the Cauchy-Riemann equations that such a function is a horizontally (weakly) CONFORMAL HARMONIC submersion. Harmonic morphisms are maps $(M,g)\rightarrow (N,h)$ between Riemannian manifolds generalizing holomorphic functions from Kähler manifolds. If the codomain is the complex plane any regular fibre is a MINIMAL submanifold of $M$ of codimension $2$. In this talk we will give a brief introduction to the theory of harmonic morphisms and then discuss the existence of complex-valued solutions from Lie groups and symmetric spaces.
