  Let $\rho_C$ be the regularity of the Hilbert function of a projective curve
$C$ in $\mathbb P^n_K$ over an algebraically closed field $K$ and
$\alpha_1,...,\alpha_{n-1}$ be minimal degrees for which there exists a
complete intersection of type $(\alpha_1,...,\alpha_{n-1})$ containing the
curve $C$. Then the Castelnuovo-Mumford regularity of $C$ is upper bounded by
$\max\{\rho_C+1,\alpha_1+...+\alpha_{n-1}-(n-2)\}$. We study and, for space
curves, refine the above bound providing several examples.
