  We study the organization and dynamics of growing directed networks. These
networks are built by adding nodes successively in such a way that each new
node has $K$ directed links to the existing ones. The organization of a growing
directed network is analyzed in terms of the number of ``descendants'' of each
node in the network. We show that the distribution $P(S)$ of the size, $S$, of
the descendant cluster is described generically by a power-law,
  $P(S) \sim S^{-\eta}$, where the exponent $\eta$ depends on the value of $K$
as well as the strength of preferential attachment. We determine that, in the
case of growing random directed networks without any preferential attachment,
$\eta$ is given by $1+1/K$. We also show that the Boolean dynamics of these
networks is stable for any value of $K$. However, with a small fraction of
reversal in the direction of the links, the dynamics of growing directed
networks appears to operate on ``the edge of chaos'' with a power-law
distribution of the cycle lengths. We suggest that the growing directed network
may serve as another paradigm for the emergence of the scale-free features in
network organization and dynamics.
