  Let $E$ be an indecomposable rank two vector bundle on the projective space
$\PP^n, n \ge 3$, over an algebraically closed field of characteristic zero. It
is well known that $E$ is arithmetically Buchsbaum if and only if $n=3$ and $E$
is a null-correlation bundle. In the present paper we establish an analogous
result for rank two indecomposable arithmetically Buchsbaum vector bundles on
the smooth quadric hypersurface $Q_n\subset\PP^{n+1}$, $n\ge 3$. We give in
fact a full classification and prove that $n$ must be at most 5. As to
$k$-Buchsbaum rank two vector bundles on $Q_3$, $k\ge2$, we prove two
boundedness results.
