Summary: The limits of a one-parameter family of spacetimes are defined, and the properties of such limits discussed. The definition is applied to an investigation of the Schwarzschild solution as a limit of the Reissner-Nordström solution as the charge parameter goes to zero. Two new techniques – rigidity of a geometrical structure and Killing transport – are introduced. Several applications of these two subjects, both to limits and to certain other questions in differential geometry, are discussed.