Most quantum states of condensed matter are classified by the symmetries they break. For example, crystalline solids break translational symmetry, and ferromagnets break rotational symmetry. By contrast, topological states of matter evade traditional symmetry-breaking classification schemes, and they signal the existence of a fundamentally different organizational principle of quantum matter. The integer and fractional quantum Hall effects were the first topological states to be discovered in the 1980s, but they exist only in the presence of large magnetic fields. The search for topological states of matter that do not require magnetic fields for their observation led to the theoretical prediction in 2006 and experimental observation in 2007 of the so-called quantum spin Hall effect in HgTe quantum wells, a new topological state of quantum matter. In this article, we review the theoretical foundations and experimental discovery of the quantum spin Hall effect.