  This report attempts a clean presentation of the theory of harmonic maps from
complex and K\"ahler manifolds to Riemannian manifolds. After reviewing the
theory of harmonic maps between Riemannian manifolds initiated by
Eells--Sampson and the Bochner technique, we specialize to K\"ahler domains and
introduce pluriharmonic maps. We prove a refined Bochner formula due to Siu and
Sampson and its main consequences, such as the strong rigidity results of Siu.
We also recount the applications to symmetric spaces of noncompact type and
their relation to Mostow rigidity. Finally, we explain the key role of this
theory for the nonabelian Hodge correspondence relating the character variety
of a compact K\"ahler manifold and the moduli space of Higgs bundles.
