Metal nanocrystals with concave surfaces are interesting for a wide variety of applications that are related to catalysis, plasmonics, and surface-enhanced spectroscopy. This interest arises from their high-index facets, surface cavities, and sharp corners/edges. Two major challenges are associated with this novel class of nanocrystals: 1) how to generate a concave surface with negative curvature, which is not favored by thermodynamics owing to its higher energy than the convex counterpart; and 2) how to stabilize the morphology of a nanocrystal with concave structures on the surface. Recently, a number of different procedures have been developed for the synthesis of noble-metal nanocrystals with concave surfaces. This Review provides a brief account of these developments, with the aim of offering new insights into the growth mechanisms. We focus on methods based on two general strategies: 1) site-specific dissolution through etching and galvanic replacement; and 2) directionally controlled overgrowth by facet-selective capping, kinetic control, and template-directed epitaxy. Their enhanced catalytic and electrocatalytic properties are also described.