  Let $J$ be an ideal of a noetherian local ring $R$. We show new results on
the set of attached primes $\Att_R(\LCMo ^l_J(R))$ of a local cohomology module
$\LCMo ^l_J(R)$. To prove our results we establish and use new relations
between the set $\Att_R(\LCMo ^l_J(R))$ of attached primes of a local
cohomology module and the set $\Ass_R(D(\LCMo ^l_J(R)))$ of associated primes
of the Matlis dual of the same local cohomology module.
