  The biplanar crossing number of a graph $G$ is the minimum number of
crossings over all possible drawings of the edges of $G$ in two disjoint
planes. We present new bounds on the biplanar crossing number of complete
graphs and complete bipartite graphs. In particular, we prove that the biplanar
crossing number of complete bipartite graphs can be approximated to within a
factor of $3$, improving over the best previously known approximation factor of
$4.03$. For complete graphs, we provide a new approximation factor of $3.17$,
improving over the best previous factor of $4.34$. We provide similar improved
approximation factors for the $k$-planar crossing number of complete graphs and
complete bipartite graphs, for any positive integer $k$. We also investigate
the relation between (ordinary) crossing number and biplanar crossing number of
general graphs in more depth, and prove that any graph with a crossing number
of at most $10$ is biplanar.
