Higher dimensional analog to planar graphs? We usually say that a graph is planar if it can be embedded into 2-space s.t. no edges intersect.  Here's a different way to describe the same situation: a graph is planar if it can be embedded into 2-space s.t. the edges form the boundaries of cells which nowhere overlap.
My question: Can you take this second definition and extend the idea to higher dimensions?  For instance, allow the cells to be volumes in 3-space - like soap bubbles where the graph edges are the intersection of soap film walls.
 A: Another generalization is along the genus axis, i.e., consider which graphs can be embedded into a 2d compact surface, such as an $n$-torus. This area is called topological graph theory.
A: A planar graph is nothing other than a triangulation of the $2$-sphere (via stereographic projection), at least if one allows arbitrary polygons instead of just triangles, so a natural generalization to three dimensions is triangulations of the $3$-sphere where one allows arbitrary (convex) polyhedra instead of just simplices. One can of course talk about triangulations of arbitrary manifolds. 
A: I'm not clear about your definition, for a graph below, what are cells and boundaries?

(source: yaroslavvb.com)
A: Another -- I think more natural -- concept is the (PL)-embeddability of $k$-dimensional simplicial complexes into $R^d$, for certain pairs $k,d$. The case $k=1$, $d=2$ is graph planarity.
There are many recent algorithmic result on this, including polynmiality in the metastable dimension range ($d\geq 3(k+1)/2$) with fixed dimensions, NP-hardness (and probably undecidability) for larger $k$, and proven undecidability for $k=d$. 
