Kappa coefficients are measures of correlation between categorical variables often used as reliability or validity coefficients. We recapitulate development and definitions of the K (categories) by M (ratings) kappas (K×M), discuss what they are well- or ill-designed to do, and summarize where kappas now stand with regard to their application in medical research. The 2×M(M⩾2) intraclass kappa seems the ideal measure of binary reliability; a 2×2 weighted kappa is an excellent choice, though not a unique one, as a validity measure. For both the intraclass and weighted kappas, we address continuing problems with kappas. There are serious problems with using the K×M intraclass (K>2) or the various K×M weighted kappas for K>2 or M>2 in any context, either because they convey incomplete and possibly misleading information, or because other approaches are preferable to their use. We illustrate the use of the recommended kappas with applications in medical research. Copyright © 2002 John Wiley & Sons, Ltd.