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