Quantum communication (QC) represents a key enabler for many quantum applications. However, low information rates and short propagation distances, limit the development of this field and its practical applications. High-dimensional (Hi-D) QC can address these challenges enhancing the information rate and the systems error tolerance. We report our recent results on Hi-D quantum communication where we prove the capability of preparing, manipulating, transmitting and measuring Hi-D quantum states through multi-core and multimode fiber.