Products of ratios of independent Gamma random variables (r.v.'s) are relevant in many tests of hypotheses. Obtaining explicit manageable expressions for their p.d.f. and c.d.f. is a challenging problem. In this monograph we take this challenge. The book tries to illustrate the use of several techniques, exhibiting a balanced blend between theory and a good number of examples. A large number of graphs and tables illustrate several particular aspects of the distributions being studied. Besides the exact distribution, we also consider near-exact ones, obtained through a new concept of approximation of the characteristic function. Computational modules are provided to implement all distributions developed. The approach followed enabled an easy extension to the non-central case and to negative power parameters, greatly widening the domain of application of the results obtained. As particular immediate cases we have the distribution of products and ratios of many known distributions, among which folded T, folded Cauchy, Beta prime or Beta second kind and, of course, F r.v.'s. The book is intended for an audience at the graduate or post-graduate level, with focus on Distribution Theory.