  For supergavrity solutions which are the product of an anti-de Sitter space
with an Einstein space X, we study the relation between the amount of
supersymmetry preserved and the geometry of X. Depending on the dimension and
the amount of supersymmetry, the following geometries for X are possible, in
addition to the maximally supersymmetric spherical geometry: Einstein-Sasaki in
dimension 2k+1, 3-Sasaki in dimension 4k+3, 7-dimensional manifolds of weak G_2
holonomy and 6-dimensional nearly Kaehler manifolds. Many new examples of such
manifolds are presented which are not homogeneous and have escaped earlier
classification efforts. String or M theory in these vacua are conjectured to be
dual to superconformal field theories. The brane solutions interpolating
between these anti-de Sitter near-horizon geometries and the product of
Minkowski space with a cone over X lead to an interpretation of the dual
superconformal field theory as the world-volume theory for branes at a conical
singularity (cone branes). We propose a description of those field theories
whose associated cones are obtained by (hyper-)Kaehler quotients.
