A directed graph is a mathematical model and data structure which formally consists of a set of vertices and a set of arcs between them. Directed graphs have a wide range of applications in computer science and mathematics, as well as in various other disciplines and industries. In this thesis, some fundamental properties for directed graphs were described, as well as algorithms used for testing directed graphs for these properties. Acyclic directed graphs contain no cycles. Computationally, acyclicity is determined by constructing a topological vertex ordering. Johnson's algorithm was described, which finds all elementary cycles in a directed graph, if they exist. Eulerian directed graphs were described, as well as Hierholzer's algorithm which finds an Eulerian circuit in a directed graph, if it exists. The problem of traversing all the vertices or arcs in a given directed graph is a common occurrence in various attempts at finding a solution to problems modeled by directed graphs. In the thesis, the depth first search algorithm was described. This method was applied to the problem of finding strongly connected components in a directed graph. The classifications of directed graphs with a certain number of vertices is still an open problem. In this thesis, the terms directed graph isomorphism and isomorphic directed graphs were described, as well as the challenges which come with trying to classify them.