  In this paper we introduce a notion of planarity for graphs that are
presented in a streaming fashion. A $\textit{streamed graph}$ is a stream of
edges $e_1,e_2,...,e_m$ on a vertex set $V$. A streamed graph is
$\omega$-$\textit{stream planar}$ with respect to a positive integer window
size $\omega$ if there exists a sequence of planar topological drawings
$\Gamma_i$ of the graphs $G_i=(V,\{e_j \mid i\leq j < i+\omega\})$ such that
the common graph $G^{i}_\cap=G_i\cap G_{i+1}$ is drawn the same in $\Gamma_i$
and in $\Gamma_{i+1}$, for $1\leq i < m-\omega$. The $\textit{Stream
Planarity}$ Problem with window size $\omega$ asks whether a given streamed
graph is $\omega$-stream planar. We also consider a generalization, where there
is an additional $\textit{backbone graph}$ whose edges have to be present
during each time step. These problems are related to several well-studied
planarity problems.
  We show that the $\textit{Stream Planarity}$ Problem is NP-complete even when
the window size is a constant and that the variant with a backbone graph is
NP-complete for all $\omega \ge 2$. On the positive side, we provide
$O(n+\omega{}m)$-time algorithms for (i) the case $\omega = 1$ and (ii) all
values of $\omega$ provided the backbone graph consists of one $2$-connected
component plus isolated vertices and no stream edge connects two isolated
vertices. Our results improve on the Hanani-Tutte-style $O((nm)^3)$-time
algorithm proposed by Schaefer [GD'14] for $\omega=1$.
