Dimension reduction for regression is a prominent issue today because technological advances now allow scientists to routinely formulate regressions in which the number of predictors is considerably larger than in the past. While several methods have been proposed to deal with such regressions, principal components (PCs) still seem to be the most widely used across the applied sciences. We give a broad overview of ideas underlying a particular class of methods for dimension reduction that includes PCs, along with an introduction to the corresponding methodology. New methods are proposed for prediction in regressions with many predictors.