Autocatalytic models for the emergence of homochirality have been of interest for more than half a century. The sole experimental example of such an amplifying autocatalytic reaction is the Soai reaction. In this review, we trace the history of the theoretical models and the experimental work that led up to the discovery of the remarkable, singular Soai reaction. The experimental and computational studies that have helped to delineate the mechanism of this reaction are discussed in detail. Studies of both the concept of chiral symmetry breaking as well as the subsequent chiral amplification process are discussed. Particular attention is paid to flaws in some of the published models, and suggestions are offered for how such issues might be avoided in future work. The outlook in the search for a prebiotically plausible version of such an amplifying autocatalytic system is presented.