  A $k$-stack layout (or $k$-page book embedding) of a graph consists of a
total order of the vertices, and a partition of the edges into $k$ sets of
non-crossing edges with respect to the vertex order. The stack number of a
graph is the minimum $k$ such that it admits a $k$-stack layout. In this paper
we study a long-standing problem regarding the stack number of planar directed
acyclic graphs (DAGs), for which the vertex order has to respect the
orientation of the edges. We investigate upper and lower bounds on the stack
number of several families of planar graphs: We improve the constant upper
bounds on the stack number of single-source and monotone outerplanar DAGs and
of outerpath DAGs, and improve the constant upper bound for upward planar
3-trees. Further, we provide computer-aided lower bounds for upward (outer-)
planar DAGs.
