  Let $G=(V,E)$ be a connected graph, let $v\in V$ be a vertex and let $e=uw\in
E$ be an edge. The distance between the vertex $v$ and the edge $e$ is given by
$d_G(e,v)=\min\{d_G(u,v),d_G(w,v)\}$. A vertex $w\in V$ distinguishes two edges
$e_1,e_2\in E$ if $d_G(w,e_1)\ne d_G(w,e_2)$. A set $S$ of vertices in a
connected graph $G$ is an edge metric generator for $G$ if every two edges of
$G$ are distinguished by some vertex of $S$. The smallest cardinality of an
edge metric generator for $G$ is called the edge metric dimension and is
denoted by $edim(G)$. In this article we introduce the concept of edge metric
dimension and initiate the study of its mathematical properties. We make a
comparison between the edge metric dimension and the standard metric dimension
of graphs while presenting some realization results concerning the edge metric
dimension and the standard metric dimension of graphs. We prove that computing
the edge metric dimension of connected graphs is NP-hard and give some
approximation results. Moreover, we present some bounds and closed formulae for
the edge metric dimension of several classes of graphs.
