  Let I=I(D) be the edge ideal of a weighted oriented graph D. We determine the
irredundant irreducible decomposition of I. Also, we characterize the
associated primes and the unmixed property of I. Furthermore, we give a
combinatorial characterization for the unmixed property of I, when D is
bipartite, D is a whisker or D is a cycle. Finally, we study the Cohen-Macaulay
property of I.
