  Given a digraph $D$, the Minimum Leaf Out-Branching problem (MinLOB) is the
problem of finding in $D$ an out-branching with the minimum possible number of
leaves, i.e., vertices of out-degree 0. We prove that MinLOB is polynomial-time
solvable for acyclic digraphs. In general, MinLOB is NP-hard and we consider
three parameterizations of MinLOB. We prove that two of them are NP-complete
for every value of the parameter, but the third one is fixed-parameter
tractable (FPT). The FPT parametrization is as follows: given a digraph $D$ of
order $n$ and a positive integral parameter $k$, check whether $D$ contains an
out-branching with at most $n-k$ leaves (and find such an out-branching if it
exists). We find a problem kernel of order $O(k^2)$ and construct an algorithm
of running time $O(2^{O(k\log k)}+n^6),$ which is an `additive' FPT algorithm.
We also consider transformations from two related problems, the minimum path
covering and the maximum internal out-tree problems into MinLOB, which imply
that some parameterizations of the two problems are FPT as well.
