A \emph{three-dimensional grid drawing} of a graph is a placement of the
vertices at distinct points with integer coordinates, such that the straight
line segments representing the edges do not cross. Our aim is to produce
three-dimensional grid drawings with small bounding box volume. We prove that
every n-vertex graph with bounded degeneracy has a three-dimensional grid
drawing with O(n3/2) volume. This is the broadest class of graphs admiting
such drawings. A three-dimensional grid drawing of a directed graph is
\emph{upward} if every arc points up in the z-direction. We prove that every
directed acyclic graph has an upward three-dimensional grid drawing with
(n3) volume, which is tight for the complete dag. The previous best upper
bound was O(n4). Our main result is that every c-colourable directed
acyclic graph (c constant) has an upward three-dimensional grid drawing with
O(n2) volume. This result matches the bound in the undirected case, and
improves the best known bound from O(n3) for many classes of directed
acyclic graphs, including planar, series parallel, and outerplanar