  In this paper, we introduce and study the multilevel-planarity testing
problem, which is a generalization of upward planarity and level planarity. Let
$G = (V, E)$ be a directed graph and let $\ell: V \to \mathcal P(\mathbb Z)$ be
a function that assigns a finite set of integers to each vertex. A
multilevel-planar drawing of $G$ is a planar drawing of $G$ such that the
$y$-coordinate of each vertex $v \in V$ is $y(v) \in \ell(v)$, and each edge is
drawn as a strictly $y$-monotone curve. We present linear-time algorithms for
testing multilevel planarity of embedded graphs with a single source and of
oriented cycles. Complementing these algorithmic results, we show that
multilevel-planarity testing is NP-complete even in very restricted cases.
