we discuss traveling front solutions $u(t,x) = u(x-ct)$ of reaction-diffusion equations $u_t = lu + f(u)$ in 1d with ignition reactions $f$ and diffusion operators $l$ generated by symmetric levy processes $x_t$. existence and uniqueness of fronts are well-known in the cases of classical diffusion (i.e., when l is the laplacian) as well as some non-local diffusion operators. we extend these results to general levy operators, showing that a weak diffusivity in the underlying process - in the sense that the first moment of $x_1$ is finite - gives rise to a unique (up to translation) traveling front. we also prove that our result is sharp, showing that no traveling front exists when the first moment of $x_1$ is infinite.