  In this paper, we use the theory of Riordan matrices to introduce the notion
of an oriented Riordan graph. The oriented Riordan graphs are a far-reaching
generalization of the well known and well studied Toeplitz oriented graphs and
tournament. The main focus in this paper is the study of structural properties
of the oriented Riordan graphs which includes a fundamental decomposition
theorem and fractal property. Finally, we introduce the generalization of the
oriented Riordan graph who is called a $p$-Riordan graph.
