This Mandelbrot Festschrift is about the development and applications of fractal geometry and the concepts of self-similarity. The reader finds out where, how and with what results fractal geometry has been applied. The balance between applications in the physical sciences and mathematics is unique to this book. It contains contributions of well-known physicists, mathematicians, material scientists, biologists, geologists, chemists, economists, astronomers and artists. Among the topics addressed are the universe, bacterial growth, stock prices, automatic sequences, number theory, percolation problems, self-organized criticality, Ising models, Levy stable processes, thermodynamics, signal analysis, catalysis, oil recovery, turbulence and earthquakes.