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In this thesis we consider background geometries\nresulting from string theory compactifications. In particular,\nwe investigate supersymmetric vacuum spaces of supergravity \ntheories and topological twisted sigma models\nby means of classical and generalised G-structures.\n\nIn the first part we compactify 11d supergravity \non seven-dimensional manifolds due to phenomenological reasons.\nA certain amount of supersymmetry forces the internal background\nto admit a classical SU(3)- or G2-structure. Especially, in the case\nthat the four-dimensional space is maximally symmetric and four form fluxes\nare present we calculate the relation to the intrinsic torsion.\n\nThe second and main part is two-fold. Firstly, we realise \nthat generalised geometries on six-dimensional manifolds are a natural \nframework to study T-duality and mirror symmetry, in particular if the\nB-field is non-vanishing. An explicit mirror map is given and we apply this \nidea to the generalised formulation of a topological twisted sigma model.\nImplications of mirror symmetry are studied, \ne.g. observables and topological A- and B-branes.\n\nSecondly, we show that seven-dimensional NS-NS backgrounds in\ntype II supergravity theories can be described by generalised \nG2-geometries. A compactification on six manifolds leads to a new structure.\nWe call this geometry a generalised SU(3)-structure. We study the\nrelation between generalised SU(3)- and G2-structures on six- and\nseven-manifolds and generalise the Hitchin-flow equations. Finally, we \nfurther develop the generalised SU(3)- and \nG2-structures via a constrained variational principle to incorporate \nalso the remaining physical R-R fields.