A number of methods for examining the residuals remaining after a conventional analysis of variance or least-squares fitting have been explored during the past few years. These give information on various questions of interest, and in particular, aid in assessing the validity or appropriateness of the conventional analysis. The purpose of this paper is to make a variety of these techniques more easily available, so that they can be tried out more widely. Techniques of analysis, some graphical, some wholly numerical, and others mixed, are discussed in terms of the residuals that result from fitting row and column means to entries in a two-way array (or in several two-way arrays). Extensions to more complex situations, and some of the uses of the results of examination, are indicated.