Mandelbrot's fractal geometry provides both a description and a mathematical model for many of the seemingly complex shapes found in nature. Such shapes often possess a remarkable invariance under changes of magnification. This statistical self-similarity may be characterized by a fractal dimension D, a number that agrees with our intuitive notion of dimension but need not be an integer. A brief mathematical characterization of random fractals is presented with emphasis on variations of Mandelbrot’s fractional Brownian motion. The important concepts of fractal dimension and exact and statisical self-similarity and self-affinity will be reviewed. The various methods and difficulties of estimating the fractal dimension and lacunarity from experimental images or point sets are summarized. © 1986 IOP Publishing Ltd.