Quantum communication, i.e., the ability to transport a quantum state from one place to another, represents a crucial task for many quantum applications, i.e. quantum cryptography, quantum secret sharing and quantum networks. However current systems present main limitations in terms of low information rates, short propagation distances and low compatibility with today classical optical infrastructure. These restrictions bound the development of this field and its practical applications. High-dimensional quantum communication can help in overcoming these challenges enhancing the information rate and the system error tolerance. We here report our recent results on high-dimensional fiber based quantum communication, both with multicore and multimode fibers, in which we prove the capability of preparing, manipulating, transmitting and measuring advanced quantum states with excellent fidelities. Our results pave the way towards high-dimensional quantum communication in an optical fiber infrastructure.