In the present thesis, some aspects of superstrings in general backgrounds are studied. The thesis divides into three parts. The first is devoted to a careful study of very convenient superspace conventions which are a basic tool for the second part. We will formulate a theorem that gives a clear statement about when the signs of a superspace calculation can be omitted. The second part describes the type II superstring using Berkovits' pure spinor formalism. Being effectively an embedding into superspace, target space supersymmetry is manifest in the formulation and coupling to general backgrounds (including Ramond-Ramond fields) is treatable. We will present a detailed derivation of the supergravity constraints as it was given already by Berkovits and Howe some years ago. The derivation will at several points differ from the original one and will use new techniques like a covariant variation principle. In addition, we will stay throughout in the Lagrangian formalism in contrast to Berkovits and Howe. Also the order in which we obtain the constraints and at some points the logic will differ. As a new result we present the explicit form of the BRST transformation of the worldsheet fields, which was before given only for the heterotic case. Having obtained all the constraints, we go one step further and derive the form of local supersymmetry transformations of the fermionic fields. This provides a contact point of the Berkovits string in general background to those supergravity calculations which derive generalized Calabi Yau conditions from effective four-dimensional supersymmetry. The mathematical background for this setting is the so-called generalized complex geometry (GCG) which is in turn the motivation for the last part.
The third and last part is based on the author's recent paper on derived brackets from sigma models which was motivated by the study of GCG. It is shown in there, how derived brackets naturally arise in sigma-models via Poisson- or antibrackets, generalizing an observation by Alekseev and Strobl. On the way to a precise formulation of this relation, an explicit coordinate expression for the derived bracket is obtained. The generalized Nijenhuis tensor of generalized complex geometry is shown to coincide up to a de-Rham closed term with the derived bracket of the structure with itself and a new coordinate expression for this tensor is presented. The insight is applied to two-dimensional sigma models in a background with generalized complex structure. The appendix contains introductions to geometric brackets and to aspects of generalized complex geometry. It further contains detailed reviews on aspects of Noether's theorem, on the Bianchi identities (including Dragon's theorem), on supergauge transformations and the WZ gauge and on important relations for Gamma -matrices (especially in ten dimensions).
A further appendix is devoted to the determination of the (super)connection starting from different torsion- or invariance constraints.