  A topological drawing of a graph is fan-planar if for each edge $e$ the edges
crossing $e$ form a star and no endpoint of $e$ is enclosed by $e$ and its
crossing edges. A fan-planar graph is a graph admitting such a drawing.
Equivalently, this can be formulated by three forbidden patterns, one of which
is the configuration where $e$ is crossed by two independent edges and the
other two where $e$ is crossed by two incident edges in a way that encloses
some endpoint of $e$. A topological drawing is simple if any two edges have at
most one point in common.
  Fan-planar graphs are a new member in the ever-growing list of topological
graphs defined by forbidden intersection patterns, such as planar graphs and
their generalizations, Tur\'an-graphs and Conway's thrackle conjecture. Hence
fan-planar graphs fall into an important field in combinatorial geometry with
applications in various areas of discrete mathematics. As every $1$-planar
graph is fan-planar and every fan-planar graph is $3$-quasiplanar, they also
fit perfectly in a recent series of work on nearly-planar graphs from the area
of graph drawing and combinatorial embeddings.
  In this paper we show that every fan-planar graph on $n$ vertices has at most
$5n-10$ edges, even though a fan-planar drawing may have a quadratic number of
crossings. Our bound, which is tight for every $n \geq 20$, indicates how
nicely fan-planar graphs fit in the row with planar graphs ($3n-6$ edges) and
$1$-planar graphs ($4n-8$ edges). With this, fan-planar graphs form the largest
non-trivial class of topological graphs defined by forbidden patterns, for
which the maximum number of edges on $n$ vertices is known exactly.
