The quantization of the Teichmueller spaces of Riemann surfaces has found important applications to conformal field theory and N=2 supersymmetric gauge theories. The aim of the talk is to construct a generalization of the quantum Teichmueller theory which describes the quantum theory of the Teichmueller spaces of Super-Riemann surfaces. One can observe that the operators in the quantum Teichmueller theory can be build combinatorially from a simple quantum group, the Borel half of U_q(sl(2)). The idea is to replace the U_q(sl(2)) algebra by a suitable quantum superalgebra, U_q(osp(1|2)). We aim to demonstrate that the resulting quantum theory is nothing else but the quantum theory of the Teichmueller spaces of Super-Riemann surfaces.