This review is concerned with the mechanical forces that cause epithelial sheets to invaginate during morphogenesis. Interest in this problem is currently increasing and a variety of models, each with a different emphasis, have been formulated to explain mechanical aspects of epithelial folding. A critical evaluation of the experimental evidence bearing on this problem leads to the following conclusions. (1) The most popular model of invagination, one based on microfilament-mediated cell shape change, should be re-examined, given the limitations of the experimental evidence usually offered in its support. Recent experiments with permeabilized epithelia offer a promising approach for confirming the validity of this model. (2) Current hypotheses based on disparities in the adhesive properties of epithelial cells are consistent with available data, but appear to be impossible to test directly at this time. (3) There is evidence that suggests that cell growth and division are involved in invagination during the branching morphogenesis of some epithelio-mesenchymal organs, but it has been shown that these processes are not involved in other cases. (4) Recent studies demonstrate that some epithelial invaginations are accompanied by movements of cells, both in the form of rearrangement (exchange of nearest neighbors) and involution (flow of surrounding cells into the invaginating region). (5) A general conclusion that may be drawn from the data now available is that several different mechanisms of epithelial folding operate during morphogenesis.