Harmonic morphisms are mappings of Riemannian manifolds which preserve solutions of Laplace's equation; elementary examples are conformal transformations of the complex plane. The concept can be traced back to work of Jacobi in 1848. For probabilists working in stochastic processes, harmonic morphisms are Brownian path-preserving transformations. Harmonic morphisms can be characterized as harmonic maps which satisfy an additional condition which is dual to the condition of weak conformality.

We shall give an overview of the subject, mentioning especially twistorial constructions, and finishing with some recent work to understand these using the bicomplex numbers.