  Let $G$ be a finite simple graph, and $J_G$ denote the binomial edge ideal of
$G$. In this article, we first compute the $\mathrm{v}$-number of binomial edge
ideals corresponding to Cohen-Macaulay closed graphs. As a consequence, we
obtain the $\mathrm{v}$-number for paths. For cycle and binary tree graphs, we
obtain a sharp upper bound for $\mathrm{v}(J_G)$ using the number of vertices
of the graph. We characterize all connected graphs $G$ with $\mathrm{v}(J_G) =
2$. We show that for a given pair $(k,m), k\leq m$, there exists a graph $G$
with an associated monomial edge ideal $I$ having $\mathrm{v}$-number equal to
$k$ and regularity $m$. If $2k \leq m$, then there exists a binomial edge ideal
with $\mathrm{v}$-number $k$ and regularity $m$. Finally, we compute
$\mathrm{v}$-number of powers of binomial edge ideals with linear resolution,
thus proving a conjecture on the $\mathrm{v}$-number of powers of a graded
ideal having linear powers, for the class of binomial edge ideals.
