  Baker and Norine proved a Riemann--Roch theorem for divisors on undirected
graphs. The notions of graph divisor theory are in duality with the notions of
the chip-firing game of Bj\"orner, Lov\'asz and Shor. We use this connection to
prove Riemann--Roch-type results on directed graphs. We give a simple proof for
a Riemann--Roch inequality on Eulerian directed graphs, improving a result of
Amini and Manjunath. We also study possibilities and impossibilities of
Riemann--Roch-type equalities in strongly connected digraphs and give examples.
We intend to make the connections of this theory to graph theoretic notions
more explicit via using the chip-firing framework.
