  In this paper, we initiate the study of global offensive $k$-alliances in
digraphs. Given a digraph $D=(V(D),A(D))$, a global offensive $k$-alliance in a
digraph $D$ is a subset $S\subseteq V(D)$ such that every vertex outside of $S$
has at least one in-neighbor from $S$ and also at least $k$ more in-neighbors
from $S$ than from outside of $S$, by assuming $k$ is an integer lying between
two minus the maximum in-degree of $D$ and the maximum in-degree of $D$. The
global offensive $k$-alliance number $\gamma_{k}^{o}(D)$ is the minimum
cardinality among all global offensive $k$-alliances in $D$. In this article we
begin the study of the global offensive $k$-alliance number of digraphs. For
instance, we prove that finding the global offensive $k$-alliance number of
digraphs $D$ is an NP-hard problem for any value $k\in
\{2-\Delta^-(D),\dots,\Delta^-(D)\}$ and that it remains NP-complete even when
restricted to bipartite digraphs when we consider the non-negative values of
$k$ given in the interval above. Based on these facts, lower bounds on
$\gamma_{k}^{o}(D)$ with characterizations of all digraphs attaining the bounds
are given in this work. We also bound this parameter for bipartite digraphs
from above. For the particular case $k=1$, an immediate result from the
definition shows that $\gamma(D)\leq \gamma_{1}^{o}(D)$ for all digraphs $D$,
in which $\gamma(D)$ stands for the domination number of $D$. We show that
these two digraph parameters are the same for some infinite families of
digraphs like rooted trees and contrafunctional digraphs. Moreover, we show
that the difference between $\gamma_{1}^{o}(D)$ and $\gamma(D)$ can be
arbitrary large for directed trees and connected functional digraphs.
