$p$-Harmonic morphisms are maps between
Riemannain manifolds that preserve solutions of $p$-
Laplace's equation. They are characterized as horizontally
weakly conformal $p$-harmonic maps so, locally, they are
solutions of an over-determined system of PDEs. I will talk
about some background of $p$-harmonic morphisms, some
calssifications and constructions of such maps, and some
applications related to minimal surfaces and biharmnonic
maps.