  Let $X(t)$, $t\geq0$, be a L\'evy process in $\mathbb{R}^d$ starting at the
origin. We study the closed convex hull $Z_s$ of $\{X(t): 0\leq t\leq s\}$. In
particular, we provide conditions for the integrability of the intrinsic
volumes of the random set $Z_s$ and find explicit expressions for their means
in the case of symmetric $\alpha$-stable L\'evy processes. If the process is
symmetric and each its one-dimensional projection is non-atomic, we establish
that the origin a.s. belongs to the interior of $Z_s$ for all $s>0$. Limit
theorems for the convex hull of L\'evy processes with normal and stable limits
are also obtained.
