  In this paper we consider colorings of oriented graphs, i.e. digraphs without
cycles of length 2. Given some oriented graph $G=(V,E)$, an oriented
$r$-coloring for $G$ is a partition of the vertex set $V$ into $r$ independent
sets, such that all the arcs between two of these sets have the same direction.
The oriented chromatic number of $G$ is the smallest integer $r$ such that $G$
permits an oriented $r$-coloring.
  In this paper we consider the Oriented Chromatic Number problem on classes of
recursively defined oriented graphs. Oriented co-graphs (short for oriented
complement reducible graphs) can be recursively defined defined from the single
vertex graph by applying the disjoint union and order composition. This
recursive structure allows to compute an optimal oriented coloring and the
oriented chromatic number in linear time. We generalize this result using the
concept of perfect orderable graphs. Therefore, we show that for acyclic
transitive digraphs every greedy coloring along a topological ordering leads to
an optimal oriented coloring. Msp-digraphs (short for minimal series-parallel
digraphs) can be defined from the single vertex graph by applying the parallel
composition and series composition. We prove an upper bound of $7$ for the
oriented chromatic number for msp-digraphs and we give an example to show that
this is bound best possible. We apply this bound and the recursive structure of
msp-digraphs to obtain a linear time solution for computing the oriented
chromatic number of msp-digraphs.
  In order to generalize the results on computing the oriented chromatic number
of special graph classes, we consider the parameterized complexity of the
Oriented Chromatic Number problem by so-called structural parameters, which are
measuring the difficulty of decomposing a graph into a special tree-structure
