  In this paper we initiate a systematic study of the Tur\'an problem for
edge-ordered graphs. A simple graph is called $\textit{edge-ordered}$, if its
edges are linearly ordered. An isomorphism between edge-ordered graphs must
respect the edge-order. A subgraph of an edge-ordered graph is itself an
edge-ordered graph with the induced edge-order. We say that an edge-ordered
graph $G$ $\textit{avoids}$ another edge-ordered graph $H$, if no subgraph of
$G$ is isomorphic to $H$.
  The $\textit{Tur\'an number}$ of an edge-ordered graph $H$ is the maximum
number of edges in an edge-ordered graph on $n$ vertices that avoids $H$. We
study this problem in general, and establish an Erd\H{o}s-Stone-Simonovits-type
theorem for edge-ordered graphs -- we discover that the relevant parameter for
the Tur\'an number of an edge-ordered graph is its $\textit{order chromatic
number}$. We establish several important properties of this parameter.
  We also study Tur\'an numbers of edge-ordered paths, star forests and the
cycle of length four. We make strong connections to Davenport-Schinzel theory,
the theory of forbidden submatrices, and show an application in Discrete
Geometry.
