Putting the quantum into magnetism might, at first sight, seem like stating the obvious; the exchange interactions leading to collective magnetic behavior are, after all, a pure quantum effect. Yet, for many phenomena in magnetism this underlying quantum nature may be safely ignored at least on the qualitative level. The investigation of magnetic systems where quantum effects play a dominant role and have to be accounted for in detail has, over the last decades, evolved to be a field of very active research. On the experimental side, major boosts have come from the discovery of high-temperature superconductivity in the mid-eighties and the increasing ability of solid state chemists to fashion magnetic systems of restricted dimensionality. While hightemperature superconductivity has raised the question of the link between the mechanism of superconductivity in the cuprates and spin fluctuations and magnetic order in one- and two-dimensional spin-1/2 antiferromagnets, the new magnetic materials have exhibited a wealth of new quantum phenomena of interest in their own. In one-dimensional systems, the universal paradigm of Luttinger liquid behavior has come to the center of interest; in all restricted geometries, the interplay of low dimension, competing interactions and strong quantum fluctuations generates, beyond the usual long range ordered states, a wealth of new states of condensed matter, such as valence bond solids, magnetic plateaux, spin liquid states or spin-Peierls states, to name but a few.