Generalizations of Planar Graphs

I guess one possible generalization could be: an $m$-dimensional stratified space (i.e. "manifold with singularities") which is embeddable in $2m$-dimensional Euclidean space. Every smooth manifold can be so embedded (by Whitney's theorem), but singularities may force the ambient dimension higher, as witnessed by the simple case $m=1$ in which "stratified space" is just a graph and the embeddability condition is just planarity.

(This was just invented on the spot - I have no idea if this is actually an interesting definition...)

There are many generalizations, but one of my favorites is "neighborhood systems": intersection graphs of systems of balls in a Euclidean space of bounded dimension, with the property that any point of the space is covered by a bounded number of balls. If the dimension is two and the number of balls covering any point is at most two, these are exactly the planar graphs (Koebe-Thurston-Andreev), they have at most a linear number of edges in any dimension, and more importantly from the point of view of divide-and-conquer algorithms they have separator theorems in any dimension (Shang-Hua Teng and others).

Planar graphs can be characterized in terms of various minor monotone graph invariants such as $\mu(G)$ of Colin de Verdière, or the recent $\sigma(G)$ of Van der Holst and Pendavingh. A graph $G$ is planar if and only if $\mu(G)$, or $\sigma(G)\leq 3$.

You can relax this to $\leq 4$, which turns out to be the flat graphs $G$, those that are linklessly embeddable in 3-space. A linkless embedding can be found in polynomial time (Van der Holst) (checking planarity is linear - Hopcroft and Tarjan). There are many connections to linear algebra, topology, and combinatorial geometry.

Also, since $\mu(G)\leq 2$ if and only if $\sigma(G)\leq 2$ if and only if $G$ is outerplanar, outerplanarity can be considered to be a natural strengthening of planarity (which goes in the opposite direction from that asked by the question).

Note: There is also a $\lambda(G)$ of Van der Holst, Laurent and Schrijver (paper) which does not characterize planarity. Instead, $\lambda(G)\leq 3$ iff $G$ does not have $K_5$ or a certain graph $V_8$ as minor.