Riemann surfaces and their supersymmetric analogues appear in string theory, conformal field theory and other subfields of mathematical physics. The study of the space of complex structures which they admit, i.e. the Teichmüller theory, and its quantisation is interesting from the perspective of e.g. perturbative string theory or AGT correspondence. I will present as an introduction the ordinary, non-supersymmetric Teichmüller theory and its quantisation, stressing the connection to representation theory of Heisenberg doubles and to Liouville field theory. Afterwards, I will show how the supersymmetric Teichmüller theory introduces a number of technical points absent in the non-supersymmetric case, and generalises the link to $mathbb{Z}_2$-graded Heisenberg doubles and $mathcal{N}=1$ supersymmetric Liouville theory.

IPhT