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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.

We discuss various descriptions of a quantum particle on noncommutative space in a (possibly non-constant) magnetic field. We have tried to present the basic facts in a unified and synthetic manner, and to clarify the relationship between various approaches and results that are scattered in the literature. 1 We dedicate these notes to the memory of Julius Wess whose scientific work was largely devoted to the study of gauge fields and whose latest interests concerned physical theories on noncommutative space. The fundamental and inspiring contributions of Julius to Theoretical Physics will always bear with us, but his great kindness, his clear and enthusiastic presentations, and his precious advice will be missed by all those who had the chance to meet him.2 Corresponding author: gieres@ipnl.in2p3.fr 3 The history of the subject has been traced back by J. Wess and is reported upon in references [2,3]. 4 We apologize to the authors whose work is not explicitly cited here and we refer to [26] for a more