Iterated function systems (i.f.ss) are introduced as a unified way of generating a broad class of fractals. These fractals are often attractors for i.f.ss and occur as the supports of probability measures associated with functional equations. The existence of certain 'p-balanced' measures for i.f.ss is established, and these measures are uniquely characterized for hyperbolic i.f.ss. The Hausdorff-Besicovitch dimension for some attractors of hyperbolic i.f.ss is estimated with the aid of p-balanced measures. What appears to be the broadest framework for the exactly computable moment theory of p-balanced measures - that of linear i.f.ss and of probabilistic mixtures of iterated Riemann surfaces - is presented. This extensively generalizes earlier work on orthogonal polynomials on Julia sets. An example is given of fractal reconstruction with the use of linear i.f.ss and moment theory.