For digraphs D and H, a homomorphism of D to H is a mapping f:V(D)domV(H) such that uvinA(D) implies f(u)f(v)inA(H). Suppose D and H are two digraphs, and ciβ(u), uinV(D), iinV(H), are nonnegative integer costs. The cost of the homomorphism f of D to H is sumuinV(D)βcf(u)β(u). The minimum cost homomorphism for a fixed digraph H, denoted by MinHOM(H), asks whether or not an input digraph D, with nonnegative integer costs ciβ(u), uinV(D), iinV(H), admits a homomorphism f to H and if it admits one, find a homomorphism of minimum cost. Our interest is in proving a dichotomy for minimum cost homomorphism problem: we would like to prove that for each digraph H, MinHOM(H) is polynomial-time solvable, or NP-hard. Gutin, Rafiey, and Yeo conjectured that such a classification exists: MinHOM(H) is polynomial time solvable if H admits a k-Min-Max ordering for some kgeq1, and it is NP-hard otherwise. For undirected graphs, the complexity of the problem is well understood; for digraphs, the situation appears to be more complex, and only partial results are known. In this thesis, we seek to verify this conjecture for ``large\u27\u27 classes of digraphs including reflexive digraphs, locally in-semicomplete digraphs, as well as some classes of particular interest such as quasi-transitive digraphs. For all classes, we exhibit a forbidden induced subgraph characterization of digraphs with k-Min-Max ordering; our characterizations imply a polynomial time test for the existence of a k-Min-Max ordering. Given these characterizations, we show that for a digraph H which does not admit a k-Min-Max ordering, the minimum cost homomorphism problem is NP-hard. This leads us to a full dichotomy classification of the complexity of minimum cost homomorphism problems for the aforementioned classes of digraphs