  A graph $H$ is {\em $p$-edge colorable} if there is a coloring $\psi: E(H)
\rightarrow \{1,2,\dots,p\}$, such that for distinct $uv, vw \in E(H)$, we have
$\psi(uv) \neq \psi(vw)$. The {\sc Maximum Edge-Colorable Subgraph} problem
takes as input a graph $G$ and integers $l$ and $p$, and the objective is to
find a subgraph $H$ of $G$ and a $p$-edge-coloring of $H$, such that $|E(H)|
\geq l$. We study the above problem from the viewpoint of Parameterized
Complexity. We obtain \FPT\ algorithms when parameterized by: $(1)$ the vertex
cover number of $G$, by using {\sc Integer Linear Programming}, and $(2)$ $l$,
a randomized algorithm via a reduction to \textsc{Rainbow Matching}, and a
deterministic algorithm by using color coding, and divide and color. With
respect to the parameters $p+k$, where $k$ is one of the following: $(1)$ the
solution size, $l$, $(2)$ the vertex cover number of $G$, and $(3)$ $l -
{\mm}(G)$, where ${\mm}(G)$ is the size of a maximum matching in $G$; we show
that the (decision version of the) problem admits a kernel with $\mathcal{O}(k
\cdot p)$ vertices. Furthermore, we show that there is no kernel of size
$\mathcal{O}(k^{1-\epsilon} \cdot f(p))$, for any $\epsilon > 0$ and computable
function $f$, unless $\NP \subseteq \CONPpoly$.
