  We classify the geometry of all supersymmetric IIB backgrounds which admit
the maximal number of $G$-invariant Killing spinors. For compact stability
subgroups $G=G_2, SU(3)$ and SU(2), the spacetime is locally isometric to a
product $X_n\times Y_{10-n}$ with $n=3,4,6$, where $X_n$ is a maximally
supersymmetric solution of a $n$-dimensional supergravity theory and $Y_{10-n}$
is a Riemannian manifold with holonomy $G$. For non-compact stability
subgroups, $G=K\ltimes\bR^8$, $K=Spin(7)$, SU(4), $Sp(2)$, $SU(2)\times SU(2)$
and $\{1\}$, the spacetime is a pp-wave propagating in an eight-dimensional
manifold with holonomy $K$. We find new supersymmetric pp-wave solutions of IIB
supergravity.
