Harmonic geometric structures: the general theory and the cases of almost complex and almost contact structures

We try and describe here the general scheme of homogeneous fibre bundles and the harmonicity equations for their sections and how it applies to geometric structures arising from a reduction of the structure group. Among G-structures, we particularly pay attention to almost complex and almost contact structures and the relevant notions of harmonic sections and harmonic maps. We then review some of the more salient properties and results in these two instances. In the last section, we detail the proof of harmonicity for the class nearly cosymplectic.