  Given finitely many connected polygonal obstacles $O_1,\dots,O_k$ in the
plane and a set $P$ of points in general position and not in any obstacle, the
{\em visibility graph} of $P$ with obstacles $O_1,\dots,O_k$ is the (geometric)
graph with vertex set $P$, where two vertices are adjacent if the straight line
segment joining them intersects no obstacle. The obstacle number of a graph $G$
is the smallest integer $k$ such that $G$ is the visibility graph of a set of
points with $k$ obstacles. If $G$ is planar, we define the planar obstacle
number of $G$ by further requiring that the visibility graph has no crossing
edges (hence that it is a planar geometric drawing of $G$). In this paper, we
prove that the maximum planar obstacle number of a planar graph of order $n$ is
$n-3$, the maximum being attained (in particular) by maximal bipartite planar
graphs. This displays a significant difference with the standard obstacle
number, as we prove that the obstacle number of every bipartite planar graph
(and more generally in the class PURE-2-DIR of intersection graphs of straight
line segments in two directions) of order at least $3$ is $1$.
