Abstract: A strong edge-coloring of a graph $G$ is a proper edge-coloring with the additional property that each color class forms an induced matching in $G$. The strong chromatic index of $G$ is the minimum $k$ for which $G$ has a strong edge-coloring using $k$ colors. Erdős and Nešetřil conjectured that every graph with maximum degree $\Delta$ has strong chromatic index at most $\frac{5}{4}\Delta^2$ if $\Delta$ is even, and at most $\frac{5}{4}\Delta^2 - \frac{1}{2}\Delta + \frac{1}{4}$ if $\Delta$ is odd. If true, both cases are best possible. In 1990, Faudree, Gyárfás, Schelp, and Tuza revised this conjecture of Erdős and Nešetřil for planar graphs with maximum degree at most 3, stating that such graphs should have strong chromatic index at most 9. We verify this conjecture, which is best possible, and extend it to loopless multigraphs. In addition to our result, I will present several unresolved conjectures and areas for further research. This is joint work with A.V. Kostochka, X. Li, W. Ruksasakchai, T. Wang, and G. Yu.