A 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. Our first main result is that every \documentclass[12pt]{minimal}
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\begin{document}$n$\end{document}-vertex graph with bounded degeneracy has a three-dimensional grid drawing with \documentclass[12pt]{minimal}
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\begin{document}$\mathcal{O}\left({n^{3/2}}\right)$\end{document} volume. This is the largest known class of graphs that have such drawings. A three-dimensional grid drawing of a directed acyclic graph (dag) is upward if every arc points up in the \documentclass[12pt]{minimal}
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\begin{document}$\textsf{z}$\end{document}-direction. We prove that every dag has an upward three-dimensional grid drawing with \documentclass[12pt]{minimal}
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\begin{document}$\mathcal{O}\left({n^3}\right)$\end{document} volume, which is tight for the complete dag. The previous best upper bound was \documentclass[12pt]{minimal}
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\begin{document}$\mathcal{O}\left({n^4}\right)$\end{document}. Our main result concerning upward drawings is that every \documentclass[12pt]{minimal}
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\begin{document}$c$\end{document}-colourable dag (\documentclass[12pt]{minimal}
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\begin{document}$c$\end{document} constant) has an upward three-dimensional grid drawing with \documentclass[12pt]{minimal}
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\begin{document}$\mathcal{O}\left({n^2}\right)$\end{document} volume. This result matches the bound in the undirected case, and improves the best known bound from \documentclass[12pt]{minimal}
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\begin{document}$\mathcal{O}\left({n^3}\right)$\end{document} for many classes of dags, including planar, series parallel, and outerplanar. Improved bounds are also obtained for tree dags. We prove a strong relationship between upward three-dimensional grid drawings, upward track layouts, and upward queue layouts. Finally, we study upward three-dimensional grid drawings with bends in the edges.
