Affine signal transformations are useful for modeling self-similar structures in fractal images and shape deformations in visual motion. In the first part of this paper a theoretical framework, called affine morphology, is developed to analyze parallel and serial superpositions of affine image transformations. Affine morphology unifies and extends translation-invariant morphological image transformations and their rotation/scaling-invariant generalizations by using action of affine groups on lattices. Several theoretical aspects of affine morphology are explored for binary images. In the second part of the paper, the affine transformations are extended to gray-level images and arbitrary signals, and affine models are developed by using a sum superposition of affine signal transformations. A solution is then given to the problem of estimating the parameters of this sum-affine model using least squares algorithms, and some applications are outlined for image and speech signal processing.