  Let $M$ be a pseudo-Riemannian spin manifold of dimension $n$ and signature
$s$ and denote by $N$ the rank of the real spinor bundle. We prove that $M$ is
locally homogeneous if it admits more than ${3/4}N$ independent Killing spinors
with the same Killing number, unless $n\equiv 1 \pmod 4$ and $s\equiv 3 \pmod
4$. We also prove that $M$ is locally homogeneous if it admits $k_+$
independent Killing spinors with Killing number $\lambda$ and $k_-$ independent
Killing spinors with Killing number $-\lambda$ such that $k_++k_->{3/2}N$,
unless $n\equiv s\equiv 3\pmod 4$. Similarly, a pseudo-Riemannian manifold with
more than ${3/4}N$ independent \emph{conformal} Killing spinors is
\emph{conformally} locally homogeneous. For (positive or negative) definite
metrics, the bounds ${3/4}N$ and ${3/2}N$ in the above results can be relaxed
to ${1/2}N$ and $N$, respectively. Furthermore, we prove that a
pseudo-Riemannnian spin manifold with more than ${3/4}N$ parallel spinors is
flat and that ${1/4}N$ parallel spinors suffice if the metric is definite.
Similarly, a Riemannnian spin manifold with more than ${3/8}N$ Killing spinors
with the Killing number $\lambda \in \bR$ has constant curvature $4\lambda^2$.
For Lorentzian or negative definite metrics the same is true with the bound
${1/2}N$. Finally, we give a classification of (not necessarily complete)
Riemannian manifolds admitting Killing spinors, which provides an inductive
construction of such manifolds.
