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Understanding supersymmetric flux vacua is essential in order to connectstring theory to observable physics. In this thesis, flux vacua are studied bymaking use of two mathematical frameworks: SU(n)-structures and generalisedcomplex geometry. Manifolds with SU(n)-structure are generalisations ofCalabi-Yau manifolds. Generalised complex geometry is a geometrical frameworkthat simultaneously generalises complex and symplectic geometry. Classes offlux vacua of type II supergravity and M-theory are given on manifolds withSU(4)-structure. The N= (1,1) type IIA vacua uplift to N=1 M-theory vacua, withfour-flux that need not be (2,2) and primitive. Explicit vacua are given onStenzel space, a non-compact Calabi-Yau. These are then generalised byconstructing families of non-CY SU(4)-structures to find vacua onnon-symplectic SU(4)-deformed Stenzel spaces. It is shown that thesupersymmetry conditions for N = (2,0) type IIB can be rephrased in thelanguage of generalised complex geometry, partially in terms of integrabilityconditions of generalised almost complex structures. This rephrasing for d=2goes beyond the calibration equations, in contrast to d=4,6 where thecalibration equations are equivalent to supersymmetry. Finally, Euclidean typeII theory is examined on SU(5)-structure manifolds, where generalised equationsare found which are necessary but not sufficient to satisfy the supersymmetryequations. Explicit classes of solutions are provided here as well. Contactwith Lorentzian physics can be made by uplifting such solutions to d=1, N=1M-theory.

On flux vacua, SU(n)-structures and generalised complex geometry