Magnetic solids should, under certain circumstances, show macroscopic quantum behavior, in which coherence exists between completely distinct magnetization states, each involving a very large number of spins (~1012 spins). This article reviews the recent work in this field, concentrating particularly on macroscopic quantum tunneling (MQT) of magnetization. The two main phenomena discussed are (a) the tunneling of magnetization in singledomain particles or grains (in which some 103−104 spins rotate together through an energy barrier), and (b) the tunneling of domain walls in films or in bulk magnets; where walls containing ~ 1010 spins may tunnel off a pinning potential, or from one pinning centre to another. Some attention is also given to the quantum nucleation of magnetization reversal in a bulk magnet, and to the quantum motion of other magnetic solitons (such as vortices). After a thorough analysis of the basic grain and wall tunneling phenomena, we continue on to a discussion of the various dissipative or “decoherence” mechanisms, which destroy the phase correlations involved in tunneling. The coupling of grain magnetization to phonons, photons, and electrons is shown to have little consequence for weaklyconducting or insulating grains. Domain walls couple to these and also to magnons and impurities or defects; the 3rd order coupling to magnons can have serious effects, but if one uses pure insulators at low temperatures, these can also be ignored. As a result, theory indicates that MQT should be visible in both grains and bulk magnets at low temperatures (at least below ~1 K). The present experimental evidence for such behavior is inconclusive, partly because few experiments have been done. We discuss these experiments, and make some suggestions for future work. It is hoped this review will stimulate such work, not only because of the fundamental interest in macroscopic quantum phenomena, but also because of the considerable scope for technological innovation.