  In this PhD thesis, we give a new geometric approach to higher Teichm\"uller
theory. In particular we construct a geometric structure on surfaces,
generalizing the complex structure, and we explore its link to Hitchin
components.
  The construction of this structure, called higher complex structure, uses the
punctual Hilbert scheme of the plane. Its moduli space admits similar
properties to Hitchin's component. Given a higher complex structure, we try to
canonically deform it to a flat connection. The space of such connections,
called "parabolic", is obtained by imitating the Atiyah--Bott reduction. It is
a space of pairs of commuting differential operators. Under some conjecture, we
establish a canonical diffeomorphism between our moduli space and Hitchin's
component.
  Finally, we generalize certain constructions, like the punctual Hilbert
scheme and the higher complex structure, to the case of a simple Lie algebra.
