TITLE: Generalizations of Planar Graphs
QUESTION [36 upvotes]: This is a follow up to Harrison's question: why planar graphs are so exceptional. I would like to ask about (and collect answers to) various notions, in graph theory and beyond graph theory (topology; algebra) that generalize the notion of planar graphs, and how properties of planar graphs extend in these wider contexts.

REPLY [2 votes]: Wagner-Fáry-Stein theorem states that each (finite simple) planar graph admits a straight-line crossing-free plane drawing. On the other hand, each graph (of at most $\frak c$ vertices) admits a straight-line crossing-free three-dimensional space drawing
(any placement of vertices with no four at one plane generates such a drawing). Recently we investigate the minimum number of planes that together cover a straight-line drawing of a graph, which turned out to be independent on dimension of the space in which the graph is drawn, provided this dimension is at least three. Also, since any straight-line and circular-arc drawing can be transformed into a circular-arc drawing by an inversion map, we investigate the minimum number of spheres covering circular arc graph drawings.