Essentially a form of targeted dimension reduction, an envelopes is a nascent construct for increasing efficiency in multivariate statistics without altering the traditional goals, sometimes producing gains equivalent to taking thousands of additional observations. This is made possible by recognizing that the data may contain unanticipated variation that is effectively immaterial to estimation or prediction. This notion leads to the central construct – an envelope – for enveloping the material variation and thereby reducing estimative variation and improving inference. Envelopes link with some standard multivariate methodology. For instance, partial least squares regression depends fundamentally on an envelope at the population level, which opens the door to pursuing envelope estimators that can significantly improve upon partial least squares predictions. We will begin with an intuitive introduction to envelopes and then describe some of their inner workings. This will be followed by a discussion of envelopes in multivariate linear regression. We will also describe how to extend the scope of envelope methods well beyond linear models. The discussion will include several small examples for illustration. Emphasis will be placed on concepts and their potential impact on data analysis. Dennis Cook is Full Professor and Director of the School of Statistics, University of Minnesota. His research areas include dimension reduction, linear and nonlinear regression, experimental design, statistical diagnostics, statistical graphics and population genetics. He is author or co-author of two text book, An Introduction to Regression Graphics and Applied Regression Including Computing and Graphics, and two research monographs, Influence and Residuals in Regression and Regression Graphics: Ideas for Studying Regressions through Graphics. He has served as Associate Editor of the Journal of the American Statistical Association, The Journal of Quality Technology, Biometrika, Journal of the Royal Statistical Society and Statistica Sinica. He is a four-time recipient of the Jack Youden Prize for Best Expository Paper in Technometrics as well as the Frank Wilcoxon Award for Best Technical Paper. He received the 2005 COPSS Fisher Lecture and Award, and is a Fellow of ASA and IMS.