  We introduce the family of $k$-gap-planar graphs for $k \geq 0$, i.e., graphs
that have a drawing in which each crossing is assigned to one of the two
involved edges and each edge is assigned at most $k$ of its crossings. This
definition is motivated by applications in edge casing, as a $k$-gap-planar
graph can be drawn crossing-free after introducing at most $k$ local gaps per
edge. We present results on the maximum density of $k$-gap-planar graphs, their
relationship to other classes of beyond-planar graphs, characterization of
$k$-gap-planar complete graphs, and the computational complexity of recognizing
$k$-gap-planar graphs.
