An operator $T$ acting on a normed space $E$ is {\em numerically hypercyclic} if, for some $(x, x^*)\in \Pi(E)$, the {\em numerical orbit} $\{x^*(T^n(x)) : n\geq 0\}$ is dense in $\mathbb{C}$. We prove that finite dimensional Banach spaces with dimension at least 2 support numerically hypercyclic operators. We also characterize the numerically hypercyclic weighted shifts on classical sequence spaces.