  Given a multigraph $G=(V,E)$, the {\em edge-coloring problem} (ECP) is to
color the edges of $G$ with the minimum number of colors so that no two
adjacent edges have the same color. This problem can be naturally formulated as
an integer program, and its linear programming relaxation is called the {\em
fractional edge-coloring problem} (FECP). In the literature, the optimal value
of ECP (resp. FECP) is called the {\em chromatic index} (resp. {\em fractional
chromatic index}) of $G$, denoted by $\chi'(G)$ (resp. $\chi^*(G)$). Let
$\Delta(G)$ be the maximum degree of $G$ and let \[\Gamma(G)=\max
\Big\{\frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm {\rm
and \hskip 2mm odd} \Big\},\] where $E(U)$ is the set of all edges of $G$ with
both ends in $U$. Clearly, $\max\{\Delta(G), \, \lceil \Gamma(G) \rceil \}$ is
a lower bound for $\chi'(G)$. As shown by Seymour, $\chi^*(G)=\max\{\Delta(G),
\, \Gamma(G)\}$. In the 1970s Goldberg and Seymour independently conjectured
that $\chi'(G) \le \max\{\Delta(G)+1, \, \lceil \Gamma(G) \rceil\}$. Over the
past four decades this conjecture, a cornerstone in modern edge-coloring, has
been a subject of extensive research, and has stimulated a significant body of
work. In this paper we present a proof of this conjecture. Our result implies
that, first, there are only two possible values for $\chi'(G)$, so an analogue
to Vizing's theorem on edge-colorings of simple graphs, a fundamental result in
graph theory, holds for multigraphs; second, although it is $NP$-hard in
general to determine $\chi'(G)$, we can approximate it within one of its true
value, and find it exactly in polynomial time when $\Gamma(G)>\Delta(G)$;
third, every multigraph $G$ satisfies $\chi'(G)-\chi^*(G) \le 1$, so FECP has a
fascinating integer rounding property.
