Quantum communication, i.e. the reliable transmission of quantum states between distant parties, has settled itself in the last decade due to its inherent connection to quantum nonlocality and ultimate communication security. It has been largely explored using two-level quantum systems, qubits, as fundamental information carriers, however, the exploitation of high-dimensional quantum state, qudits, offers major advantages in quantum communication with respect to qubits. Above all, high-dimensional quantum states own the great benefit of the increased information capacity per quantum system, thus allowing for higher information rates. Qudits generation and manipulation have been largely improved over the years, but their faithful fiber transmission constitutes still the main challenge. During the three years of my Ph.D., we have investigated the fiber transmission of d-level quantum states encoded in the orbital angular momentum (OAM) of light, the emblematic degree of freedom to devise high-dimensional states. Fiber transmission of such states required the exploitation of a novel ring structured optical fiber, namely, an air-core fiber. By coupling OAM encoded qudits to such a new fiber, we have investigated both their propagation and their application in practical quantum communication protocols, but also we have inspected the robustness of quantum correlations shared among a photon pair after the transmission of one of them.