The use of fractals in fields such as molecular biology, epidemiology, landscape, ecology, geology, physics, etc., is becoming more common. In order to use fractals to model many phenomena, the researcher requires the knowledge of the fractal, or Hausdorff-Besicovitch, dimension. However, no statistical properties of the usual estimator, the entropy estimator, are known. In addition, the entropy estimator is biased high when an inefficient net is used. This dissertation develops a new estimator, the relative entropy estimator, which is asymptotically unbiased and is consistent. The estimator is asymptotically normal, and asymptotic confidence intervals are presented. An estimate of the variance of the estimator is given which does not depend on the dimension, or its estimate, using an occupancy model. The exact distribution of the estimator is also derived. Applications of the theory to various fields are presented. For example, I find that from the point of view of dimension, the logarithms of stock prices behave consistently with the classical Brownian function. Also, the relative entropy estimator gives a more realistic estimate of the dimension of surface terrain than an ad hoc estimate found in the literature. The Hausdorff dimensions of nursery-grown tree roots were estimated, and it was found that the dimension is related to the probability of the tree’s survival when the tree is planted in the wild. The dimensions of Julia sets and of the Hénon attractor were also investigated. A computer program for calculating the estimates is included. / Ph. D.