An injective k-edge-coloring of a graph G is an assignment of colors, i.e. integers in {1,. .. , k}, to the edges of G such that any two edges each incident with one distinct endpoint of a third edge, receive distinct colors. The problem of determining whether such a k-coloring exists is called Injective k-Edge-Coloring. We show that Injective 3-Edge-Coloring is NP-complete, even for triangle-free cubic graphs, planar subcubic graphs of arbitrarily large girth, and planar bipartite subcubic graphs of girth 6. Injective 4-Edge-Coloring remains NP-complete for cubic graphs. For any k ≥ 45, we show that Injective k-Edge-Coloring remains NP-complete even for graphs of maximum degree at most 5 √ 3k. In contrast with these negative results, we show that Injective k-Edge-Coloring is linear-time solvable on graphs of bounded treewidth. Moreover, we show that all planar bipartite subcubic graphs of girth at least 16 are injectively 3-edge-colorable. In addition, any graph of maximum degree at most k/2 is injectively k-edge-colorable.