Edge coloring is one of the most fundamental graph problems. A proper edge coloring of a graph is an assignment of colors to the edges of the graph such that no two adjacent edges get the same color and the minimum number of colors required for a proper edge coloring of a graph is called as chromatic index of the graph. Vizing’s theorem says that the chromatic index of a graph is either \(\varDelta \) or \(\varDelta +1\) , where \(\varDelta \) is the maximum degree of the graph. Given a graph G, the Chromatic Index problem asks if the chromatic index of G is \(\varDelta \) . This problem is known to be NP-complete even for 3-regular graphs, which makes this problem unlikely to be in FPT (unless P \(=\) NP) when parameterized by \(\varDelta (G)\) or by the clique number of the graph. In this paper, we initiate the study of the parameterized complexity of the problem. We design an FPT algorithm using dynamic programming on a given (nice) tree decomposition of the graph with respect to the treewidth parameter. This algorithm matches the running time of the best known algorithm for the problem in partial k-trees (graphs with treewidth at most k) given by X. Zhou et.al. [J. Algorithms, 1996], which uses a parallel algorithm for Chromatic Index as a subroutine. This result follows that the Chromatic Index problem is FPT for chordal graphs, when parameterized by the clique number of the graph. Moreover, we show that the problem does not admit a polynomial kernel when parameterized by the treewidth parameter unless PH collapses. On the positive side, we design a quadratic kernel parameterized by the size of a given feedback vertex set. We also design a polynomial kernel for the problem with respect to the parameters cluster vertex deletion number and clique number.