There is a recent generalization to $k$-uniform hypergraphs that are embeddable in $\mathbb{R}^d$ without edge intersections. "For $k=d=2$ the problem specializes to graph planarity":

Carl Georg Heise, Konstantinos Panagiotou, Oleg Pikhurko, Anusch Taraz. "Coloring $d$-embeddable $k$-uniform Hypergraphs." Discrete & Computational Geometry (2014) 52:663-679. (arXiv abs link.)

"We say that $H = (V, E)$ is a $k$-uniform hypergraph if the vertex set $V$ is a finite set and the edge set $E$ consists of $k$-element subsets of $V$."

There is a recent generalization to $k$-uniform hypergraphs that are embeddable in $\mathbb{R}^d$ without edge intersections. "For $k=d=2$ the problem specializes to graph planarity":

Carl Georg Heise, Konstantinos Panagiotou, Oleg Pikhurko, Anusch Taraz. "Coloring $d$-embeddable $k$-uniform Hypergraphs." Discrete & Computational Geometry (2014) 52:663-679. (arXiv abs link.)

"We say that $H = (V, E)$ is a $k$-uniform hypergraph if the vertex set $V$ is a finite set and the edge set $E$ consists of $k$-element subsets of $V$."

There is a recent generalization to $k$-uniform hypergraphs that are embeddable in $\mathbb{R}^d$ without edge intersections. "For $k=d=2$ the problem specializes to graph planarity":

Carl Georg Heise, Konstantinos Panagiotou, Oleg Pikhurko, Anusch Taraz. "Coloring $k$-embeddable $k$-uniform Hypergraphs." Discrete & Computational Geometry (2014) 52:663-679. (arXiv abs link.)

"We say that $H = (V, E)$ is a $k$-uniform hypergraph if the vertex set $V$ is a finite set and the edge set $E$ consists of $k$-element subsets of $V$."

There is a recent generalization to $k$-uniform hypergraphs that are embeddable in $\mathbb{R}^d$ without edge intersections. "For $k=d=2$ the problem specializes to graph planarity":

Carl Georg Heise, Konstantinos Panagiotou, Oleg Pikhurko, Anusch Taraz. "Coloring $d$-embeddable $k$-uniform Hypergraphs." Discrete & Computational Geometry (2014) 52:663-679. (arXiv abs link.)

"We say that $H = (V, E)$ is a $k$-uniform hypergraph if the vertex set $V$ is a finite set and the edge set $E$ consists of $k$-element subsets of $V$."