  Pluriharmonic maps form an important class of harmonic maps which includes
holomorphic maps. We study their morphisms, in particular the
inter-relationships between $(1,1)$-geodesic, pluriharmonic and
$\pm$holomorphic maps. Then we characterise pluriharmonic morphisms between
Hermitian manifolds. We make a special study of the situation where the target
is K{\"a}hler, pluriharmonic morphisms being particularly well understood for
this case.
