  We introduce and analyze a new geometric structure on topological surfaces
generalizing the complex structure. To define this so called higher complex
structure we use the punctual Hilbert scheme of the plane. The moduli space of
higher complex structures is defined and is shown to be a generalization of the
classical Teichm\"uller space. We give arguments for the conjectural
isomorphism between the moduli space of higher complex structures and Hitchin's
component.
