  Given a directed graph $G = (V, E)$, the $k$-path partition problem is to
find a minimum collection of vertex-disjoint directed paths each of order at
most $k$ to cover all the vertices of $V$. The problem has various applications
in facility location, network monitoring, transportation and others. Its
special case on undirected graphs has received much attention recently, but the
general directed version is seemingly untouched in the literature. We present
the first $k/2$-approximation algorithm, for any $k \ge 3$, based on a novel
concept of augmenting path to minimize the number of singletons in the
partition. When $k \ge 7$, we present an improved $(k+2)/3$-approximation
algorithm based on the maximum path-cycle cover followed by a careful $2$-cycle
elimination process. When $k = 3$, we define the second novel kind of
augmenting paths and propose an improved $13/9$-approximation algorithm.
