Conformally invariant couplings of two-dimensional field theories to gravity can be formulated either in a Riemannian surface or manifold (metric) framework. After a detailed presentation of the Riemannian surface approach (and comparison with the metric formalism), the authors develop its supersymmetric generalization. Super Beltrami differentials are introduced without any reference to metric (vielbein) structures and superconformal models are studied in this framework. The main goal of the work consists in the construction of local (and supersymmetric) field theories defined on arbitrary (super-)Riemann surfaces and exhibiting the (super-)holomorphic factorization in a manifest way.