A packing of graphs $G_1$ and $G_2$, both on $n$ vertices, is a set $\{H_1,H_2\}$ such that $H_1\cong G_1$, $H_2\cong G_2$, and $H_1$ and $H_2$ are edge disjoint subgraphs of $K_n$. In 1978, Sauer and Spencer [J. Combin. Theory Ser. B, 25 (1978), pp. 295--302] proved that if $|E(G_1)|+|E(G_2)|<\frac{3}{2}n-1$, then there is a packing of $G_1$ and $G_2$. Independently, Bollobás and Eldridge [J. Combin. Theory Ser. B, 25 (1978), pp. 105--124] obtained a stronger result. Namely, they proved that if $|E(G_1)|+|E(G_2)|\leq 2n-4$, then there is a packing of $G_1$ and $G_2$, provided that $\Delta(G_1)

One of the classical results in packing theory states that every graph of order n and size less than or equal to $n-2$ is packable in its complement. Moreover, the bound is sharp because the star is not packable. A similar problem arises for digraphs, namely, to find the maximal number $f_D(n)$ such that every digraph of order n and size less than or equal to $f_D(n)$ is packable. So far it is known that $\frac{7}{4}n-81\leq f_D(n)\leq2n-3$, where the upper bound is sharp. In this paper we prove that $f_D(n)=2n-o(n)$.