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The $2$-layer drawing model is a well-established paradigm to visualizebipartite graphs. Several beyond-planar graph classes have been studied underthis model. Surprisingly, however, the fundamental class of $k$-planar graphshas been considered only for $k=1$ in this context. We provide severalcontributions that address this gap in the literature. First, we show tightdensity bounds for the classes of $2$-layer $k$-planar graphs with$k\in\{2,3,4,5\}$. Based on these results, we provide a Crossing Lemma for$2$-layer $k$-planar graphs, which then implies a general density bound for$2$-layer $k$-planar graphs. We prove this bound to be almost optimal with acorresponding lower bound construction. Finally, we study relationships between$k$-planarity and $h$-quasiplanarity in the $2$-layer model and show that$2$-layer $k$-planar graphs have pathwidth at most $k+1$.

$2$-Layer $k$-Planar Graphs: Density, Crossing Lemma, Relationships, and Pathwidth