  In this paper, we analyze embeddings of grid graphs on orientable surfaces.
We determine the genus of a large class of k-dimensional grid graphs and
effective two-sided bounds for the genus of any 3-dimensional grid graph, both
in terms of a grid graph's combinatorics. As an application, we provide a
complete classification of planar and toroidal grid graphs. Our work requires a
variety of combinatorial arguments to determine effective lower bounds on the
genus of a grid graph, along with explicitly constructing embeddings of grid
graphs on surfaces to determine effective upper bounds on their genera.
