Fractal geometry is a field for researchers from different disciplines such as mathematics, physics, chemistry, biology and medicine, and is used to control complicated natural and technical phenomena. The most convincing models contain an element of randomness so that the combination of fractal geometry and stochastics arises in a canonical way. This text is concerned with the mathematical theory used at the borderline between these two fields. The contributions - from the main speakers attending the conference, "Fractal Geometry and Stochastics", held at Finsterbergen, Germany, in June 1994 - aim to highlight the principle directions of research in this area, covering such topics as: fractal sets and measures; iterated function systems; random fractals; fractals and dynamical systems; and harmonic analysis on fractals.