  In this article, we obtain an upper bound for the Castelnuovo-Mumford
regularity of powers of an ideal generated by a homogeneous quadratic sequence
in a polynomial ring in terms of the regularity of its related ideals and
degrees of its generators. As a consequence, we compute upper bounds for the
regularity of powers of several binomial ideals. We generalize a result of
Matsuda and Murai to show that the regularity of $J^s_G$ is bounded below by
$2s+\ell(G)-1$ for all $s \geq 1$, where $J_G$ denotes the binomial edge ideal
of a graph $G$ and $\ell(G)$ is the length of a longest induced path in $G$. We
compute the regularity of powers of binomial edge ideals of cycle graphs, star
graphs, and balloon graphs explicitly. Also, we give sharp bounds for the
regularity of powers of almost complete intersection binomial edge ideals and
parity binomial edge ideals.
