Quantum Shannon theory (QST) comprises various asymptotic quantum information proc essing problems, such as quantum compression, quantum channel coding, entanglement assisted communication and entanglement distillation. We adopt a view of QST as a quantitative theory of interconversion between information processing resources. In this formalism, each coding theorem corresponds to an _inequality_ between resources. This talk reviews the recent progress in QST, culminating in entropic solutions to the problem of interconverting noiseless one-way quantum communication, classical communication and entanglement in the presence of a noisy bipartite state or channel, thus unifying a large class of previously known results in the field. We shall also see classical and quantum examples of noisy resource _equalities_ involving feedback channels.