The relationship between a Noetherian local ring (R, m) and its m-adic completion can be surprisingly pathological. While the completion of a regular local ring is always a regular local ring, Nagata gave an example of a normal local ring with non-reduced completion. As Heitmann showed, there are even unique factorization domains with this property. These rings are quite hard to describe, however, and for good reason! Essentially any local ring that one can write down will be excellent, meaning that properties like reducedness and normality are shared by the ring and its completion. We discuss a set of highly non-constructive techniques, developed my Heitmann and others, to study the fine details of the relationship between a ring and its completion, allowing us to answer questions such as ``which rings are the completion of an integral domain?'' Lastly, we mention a conjectural application of these techniques to the deformation of perfectoid purity in Gorenstein rings.