  We establish properties of a new type of fractal which has partial self
similarity at all scales. For any collection of iterated functions systems with
an associated probability distribution and any positive integer V there is a
corresponding class of V-variable fractal sets or measures. These V-variable
fractals can be obtained from the points on the attractor of a single
deterministic iterated function system. Existence, uniqueness and approximation
results are established under average contractive assumptions. We also obtain
extensions of some basic results concerning iterated function systems.
