Around 1990 Hitchin proved that there is a connected component, of the representation variety of the fundamental group of a closed surface into PSL(n,R), which is homoemorphic to a ball. In many ways this Hitchin component seems to be a “higher” analogue of Teichmueller space. Similar higher analogues of Teichmueller space exist for every semisimple real Lie group G which is either a split real form or of Hermitian type. It is known that representations in these higher Teichmueller spaces have many nice properties. In this talk I will adress the question to what extent these representation arise as holonomy representations of geometric structures. I will explain in particular that the Hitchin component for PSL(4,R) can be interpreted as the moduli space of special projective structures on the unit tangent bundle of the surface. This is joint work with Olivier Guichard.