Fractal geometry can help us to describe the shapes in nature (e.g., ferns, trees, seashells, rivers, mountains) exceeding the limits imposed by Euclidean geometry. Fractal geometry is quite young: The first studies are the works by the French mathematicians Pierre Fatou (1878-1929) and Gaston Julia (1893-1978) at the beginning of the 20th century. However, only with the mathematical power of computers has it become possible to realize connections between fractal geometry and other disciplines. It is applied in various fields now, from biology to economy. Important applications also appear in computer science because fractal geometry permits us to compress images, and to reproduce, in virtual reality environments, the complex patterns and irregular forms present in nature using simple iterative algorithms executed by computers. Recent studies apply this geometry to controlling traffic in computer networks (LANs, MANs, WANs, and the Internet). The aim of this chapter is to present fractal geometry, its properties (e.g., self-similarity), and their applications in computer science.