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