Let A and B be attractors of two point-fibred iterated function systems with coding maps f and g. A transformation from A into B can be constructed by composing a section of the inverse off with g. I will outline the shape of the theory of such transformations, which are termedfractal because their graphs are typically of noninteger dimension. I will also describe the remarkable geometry of these transformationswhen the iterated functions systems are projective. I will demonstrate that (i) they can beused to provide new insights into dynamical systems; (ii) they can be used to manipulate,filter, process and efficiently store digital images; and (iii) they can be used in the visualarts. This lecture will be accessible to a general mathematically literate audience and willinclude some computer graphics. Most of the results described in this talk are new(within the last three months) and include results of joint work with Brendan Harding(ANU) and Konstantin Igudesman (Kazan State).