Techniques for facial expression generation are employed in several applications in computer graphics as well as in the processing of image and video sequences containing faces. Video coding standards such as MPEG-4 support facial expression animation. There are a number of facial expression representations that are application dependent or facial animation standard dependent and most of them require a lot of computational effort. We have developed a completely novel and effective method for representing the primary facial expressions using a model-independent set of deformation parameters (derived using rubber-sheet transformations), which can be easily applied to transform facial feature points. The developed mathematical model captures the necessary non-linear characteristics of deformations of facial muscle regions; producing well-recognizable expressions on images, sketches, and three dimensional models of faces. To show the effectiveness of the method, we developed a variety of novel applications such as facial expression recognition, expression mapping, facial animation and caricature generation.