  We investigate Riemann-Roch theory for directed graphs. The Riemann-Roch
criteria of Amini and Manjunath is generalized to all integer lattices
orthogonal to some positive vector. Using generalized notions of a
$v_0$-reduced divisor and Dhar's algorithm we investigate two chip-firing games
coming from the rows and columns of the Laplacian of a strongly connected
directed graph. We discuss how the "column" chip-firing game is related to
directed $\vec{G}$-parking functions and the "row" chip-firing game is related
to the sandpile model. We conclude with a discussion of arithmetical graphs,
which after a simple transformation may be viewed as a special class of
directed graphs which will always have the Riemann-Roch property for the column
chip-firing game. Examples of arithmetical graphs are provided which
demonstrate that either, both, or neither of the two Riemann-Roch conditions
may be satisfied for the row chip-firing game.
