Edge-distance-regularity is a concept recently introduced by the authors which issimilar to that of distance-regularity, but now the graph is seen from each of its edgesinstead of from its vertices. More precisely, a graph Γ with adjacency matrix A is edge-distance-regular when it is distance-regular around each of its edges and withthe same intersection numbers for any edge taken as a root. In this paper we studythis concept, give some of its properties, such as the regularity of Γ, and derive somecharacterizations. In particular, it is shown that a graph is edge-distance-regular if and only if its k-incidence matrix is a polynomial of degree k in A multiplied by the(standard) incidence matrix. Also, the analogue of the spectral excess theorem fordistance-regular graphs is proved, so giving a quasi-spectral characterization of edgedistance-regularity. Finally, it is shown that every nonbipartite graph which is both distance-regular and edge-distance-regular is a generalized odd graph.