In this paper we initiate a systematic study of the Turán 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 $\textit{Turán number}$ 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 Erdős-Stone-Simonovits-type theorem for edge-ordered graphs -- we discover that the relevant parameter for the Turán number of an edge-ordered graph is its $\textit{order chromatic number}$. We establish several important properties of this parameter. We also study Turán 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.