  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 $O(n^{3/2})$ 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
$(n^3)$ volume, which is tight for the complete dag. The previous best upper
bound was $O(n^4)$. Our main result is that every $c$-colourable directed
acyclic graph ($c$ constant) has an upward three-dimensional grid drawing with
$O(n^2)$ volume. This result matches the bound in the undirected case, and
improves the best known bound from $O(n^3)$ for many classes of directed
acyclic graphs, including planar, series parallel, and outerplanar.
