Fractals have a wide variety of applications in industries such as architecture and video games, as well as in research fields including soil mechanics and antenna design, amongst others. Finite collections of contractive set-valued affine functions, called iterated function systems (IFSs), have been shown to possess unique, globally attractive fixed points, termed attractors. These attractors often display fractal features. In this thesis we extend the theory of IFSs to include functions with bounded derivatives, and piecewise functions, both of which produce interesting results. Piecewise IFSs also allow for the development of fractal splicing. Additionally, we approximate a solution to a long-standing fractal inverse problem: Given the image of an attractor, what are the parameters of the IFS that produced it? We make use of neural networks to approximate this mapping.