Quantum key distribution is a promising candidate to ensure secure communication in the future. Higher-dimensional state spaces as provided by the orbital angular momentum of photons (OAM) could further improve the security and robustness to noise. When transmitted through free-space, these photons experience atmospheric turbulence which introduces distortions to the encoded OAM. A technologoy called adaptive optics has been developed in astronomy to counter-act turbulence-induced distortion. While the application of adaptive optics has been considered in the context of classical OAM communication, this thesis presents the first results for OAM-based quantum key distribution. The main result of this thesis is that the losses of the quantum mechanical properties of OAM photons in atmospheric can be mitigated by adaptive optics. Both the entanglement and the trace of the final state's density matrix can be significantly improved. Furthermore, the security of the quantum key distribution protocol can be ensured for two to three times stronger turbulence as measured by the Fried parameter. The presented results demonstrate the great potential of adaptive optics for OAM quantum cryptography.