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The first part of this paper surveys several characterizations ofTeichm\"uller space as a subset of the space of representation of thefundamental group of a surface into PSL(2,R). Special emphasis is put on(bounded) cohomological invariants which generalize when PSL(2,R) is replacedby a Lie group of Hermitian type. The second part discusses underlyingstructures of the two families of higher Teichm\"uller spaces, namely the spaceof maximal representations for Lie groups of Hermitian type and the space ofHitchin representations or positive representations for split real simple Liegroups.

Higher Teichm\"uller Spaces: from SL(2,R) to other Lie groups