  We prove a conjectured upper bound for the Castelnuovo-Mumford regularity of
binomial edge ideals of graphs, due to Matsuda and Murai. Indeed, we prove that
$\mathrm{reg}(J_G)\leq n-1$ for any graph $G$ with $n$ vertices, which is not a
path. Moreover, we study the behavior of the regularity of binomial edge ideals
under the join product of graphs.
