In recent years, there has been a rising interest in high-dimensional quantum states and their impact on quantum communication. Indeed, the availability of an enlarged Hilbert space offers multiple advantages, from larger information capacity and increased noise resilience, to novel fundamental research possibilities in quantum physics. Multiple photonic degrees of freedom have been explored to generate high-dimensional quantum states, both with bulk optics and integrated photonics. Furthermore, these quantum states have been propagated through various channels, for example, free-space links, single-mode, multicore, and multimode fibers, and also aquatic channels, experimentally demonstrating the theoretical advantages over 2D systems. Here, the state-of-the-art on the generation, propagation, and detection of high-dimensional quantum states is reviewed. Quantum communication with states living in d-dimensional Hilbert spaces, qudits, yields great benefits. However, qudits generation, transmission, and detection is not a simple task to accomplish. This review presents the state-of-the-art on the generation, propagation, and measurement of high-dimensional quantum states, highlighting their advantages, issues, and future perspectives.