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We study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in R 3 . We show that every graph is realizable as a polyhedral surface with arbitrary polygonal cells, and that this is not true if we require the cells to be convex. In particular, if the given graph contains K 5 , K 5 , 81 , or any nonplanar 3-tree as a subgraph, no such realization exists. On the other hand, all planar graphs, K 4 , 4 , and K 3 , 5 can be realized with convex cells. The same holds for any subdivision of any graph where each edge is subdivided at least once, and, by a result from McMullen et al. (Isr. J. Math. 46(1–2), 127–144 (1983)), for any hypercube. Our results have implications on the maximum density of graphs describing polyhedral surfaces with convex cells: The realizability of hypercubes shows that the maximum number of edges over all realizable n-vertex graphs is in Ω (n log n) . From the non-realizability of K 5 , 81 , we obtain that any realizable n-vertex graph has O (n 9 / 5) edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense. [ABSTRACT FROM AUTHOR]