Tunnel ionization, the fundamental process in strong field physics and attosecond science, along with the subsequent electron dynamics are typically governed by the polarization and carrier envelope phase of the incident laser pulse. Moreover, most light-matter interactions involve Gaussian beams and rely primarily on dipole-active transitions. In this article, we reveal that Orbital Angular Momentum (OAM) carrying beams enable to control tunnel ionization in atoms and molecules. The ionization process is manipulated by the sign and value of the OAM and by displacing the phase singularity. We show that the helical phase and field gradients inherent in the higher-order multipole expansion of the tunneling process cause ionization to depend on OAM. Simulations indicate that, in contrast to Gaussian beams, the ponderomotive effects can also be controlled with OAM and the asymmetry in the optical vortex. Our findings have an impact on attosecond science, spectroscopy, and super-resolution microscopy.