  We study the parameterized complexity of the directed variant of the
classical {\sc Steiner Tree} problem on various classes of directed sparse
graphs. While the parameterized complexity of {\sc Steiner Tree} parameterized
by the number of terminals is well understood, not much is known about the
parameterization by the number of non-terminals in the solution tree. All that
is known for this parameterization is that both the directed and the undirected
versions are W[2]-hard on general graphs, and hence unlikely to be fixed
parameter tractable FPT. The undirected {\sc Steiner Tree} problem becomes FPT
when restricted to sparse classes of graphs such as planar graphs, but the
techniques used to show this result break down on directed planar graphs.
  In this article we precisely chart the tractability border for {\sc Directed
Steiner Tree} (DST) on sparse graphs parameterized by the number of
non-terminals in the solution tree. Specifically, we show that the problem is
fixed parameter tractable on graphs excluding a topological minor, but becomes
W[2]-hard on graphs of degeneracy 2. On the other hand we show that if the
subgraph induced by the terminals is required to be acyclic then the problem
becomes FPT on graphs of bounded degeneracy.
  We further show that our algorithm achieves the best possible running time
dependence on the solution size and degeneracy of the input graph, under
standard complexity theoretic assumptions. Using the ideas developed for DST,
we also obtain improved algorithms for {\sc Dominating Set} on sparse
undirected graphs. These algorithms are asymptotically optimal.
