A comprehensive survey of the use of the Liouville (and super-Liouville) equation in (super)string theory outside the critical dimension, and of the complementary approach based on the discretized space-time - known as the matrix model approach. The authors pay particular attention to supersymmetry, both in the continuum formulation and through the consideration of the super-eigenvalue problem. The methods presented here are important in a large number of complex problems, e.g. random surfaces, 2-D gravity and large-N quantum chromodynamics, and this comparitive study of the different methods permits a cross-evaluation of the results when both methods are valid, combined with new predictions when only one of the methods may be applied.

Correlation Functions in the Bosonic Theory (Continuum Approach for Spherical Topology).- Hermitian Matrix Model.- Conformal Basis for Scaling Operators.- Correlation Functions for the N=1 Super Liouville Theory.- N=1 Super Eigenvalue Model.- Correlation Functions in N=2 Super Liouville Theory.- Final Remarks and Outlook.