This research begins with reaction-diffusion, first proposed by Alan Turing in 1952 to account for morphogenesis -- the formation of hydranth tentacles, leopard spots, zebra stripes, etc. Reaction-diffusion systems have been researched primarily by biologists working on theories of natural pattern formation and by chemists modeling dynamics of oscillating reactions. The past few years have seen a new interest in reaction-diffusion spring up within the computer graphics and image processing communities. However, RD s are generally unbounded, making them impractical for many applications. In this thesis we introduce a bounded and more flexible non-linear system, the ``M-lattice'', which preserves the natural pattern-formation properties of reaction-diffusion. On the theoretical front, we establish relationships between RD s and paradigms in linear systems theory and certain types of artificial ``neurally-inspired'' systems. The M-lattice is closely related to the analog Hopfield network and the CNN, but has more flexibility in how its variables interact. The bounded M-lattice enables computer or analog VLSI implementations to serve as simulation ``engines'' for a wide variety of systems of partial and ordinary differential equations. On the practical front, we have developed new applications of reaction-diffusion (formulated as the new M-lattice). These include the synthesis of visual and sound textures, restoration and enhancement of fingerprints, non-linear programming, and digital halftoning of images. Halftones were synthesized in the creatively hand-drawn ``special-effects'' style of the Wall Street Journal portraits as well as in the ``faithful-rendition'' style of error-diffusion.