  In this thesis, we investigate all warped AdS$_4$ and AdS$_3$ backgrounds
with the most general allowed fluxes that preserve more than 16 supersymmetries
in 10- and 11-dimensional supergravities. Assuming either that the internal
manifold is compact without boundary or that the isometry algebra of the
background decomposes into that of AdS and that of the transverse space, we
find that there are no AdS$_4$ backgrounds in IIB supergravity. Similarly, we
find a unique such background with 24 supersymmetries in IIA supergravity,
locally isometric to $AdS_4\times \mathbb{CP}^3$. In 11-dimensional
supergravity all more than half BPS AdS backgrounds are shown to be locally
isometric to the maximally supersymmetric $AdS_4\times S^7$ solution.
Furthermore, we prove a non-existence theorem for AdS$_3$ solutions preserving
more than 16 supersymmetries. Finally, we demonstrate that warped Minkowski
space backgrounds of the form $\mathbb{R}^{n-1,1}\times_w M^{D-n}$ ($n\geq
3,D=10,11$) in 11-dimensional and type II supergravities preserving strictly
more than 16 supersymmetries and with fields, which may not be smooth
everywhere, are locally isometric to the Minkowski vacuum $\mathbb{R}^{D-1,1}$.
In particular, all such flux compactification vacua of these theories have the
same local geometry as the maximally supersymmetric vacuum
$\mathbb{R}^{n-1,1}\times T^{d-n}$.
