||In this thesis I introduce the concepts of arc ideals, unmixed digraphs, and Cohen-Macaulay digraphs, and study their algebraic and combinatorial properties by using commutative algebra and graph theory. The main results are the complete characterizations of the unmixed digraphs with δ > 0 and Cohen-Macaulay digraphs. The thesis consists of five chapters, which are organized as follows. Chapter 1 provides the algebraic backgrounds which are needed in the thesis with the aim of introducing the Cohen-Macaulay rings and Cohen-Macaulay modules. Chapter 2 presents the basic terminology and notion for graphs and digraphs, a short exposition of matching theory, and some relevant results on two types of digraphs: acyclic digraphs and transitive digraphs. Chapter 3 mainly includes the theory of Stanley-Reisner rings, combinatorial topological characterization of Cohen-Macaulay complexes, and the properties of the Hibi ideals. Chapter 4 and Chapter 5 are dedicated to the applications of the previous chapters to the study of digraphs, in which unmixed digraphs and Cohen-Macaulay digraphs are characterized.