Density of a semigroup in a Banach space

We study conditions on a set $M$ in a Banach space $X$ which are necessary or sufficient for the set $R(M)$ of all sums $x_1+\dots+x_n$, $x_k\in M$, to be dense in $X$. We distinguish conditions under which the closure $\overline{R(M)}$ is an additive subgroup of $X$, and conditions under which this additive subgroup is dense in $X$. In particular, we prove that if $M$ is a closed rectifiable curve in a uniformly convex and uniformly smooth Banach space $X$, and does not lie in a closed half-space $\{x\in X\colon f(x)\geqslant0\}$, $f\in X^*$, and is minimal in the sense that every proper subarc of $M$ lies in an open half-space $\{x\in X\colon f(x)>0\}$, then $\overline{R(M)}=X$. We apply our results to questions of approximation in various function spaces.