This book presents a comprehensive study of Latin and hyper-Greek-Latin cubes and hypercubes, aimed at uncovering the impact of their combinatorial properties on the evaluation of parametric functions and the testing of statistical hypotheses. It explores how various factors, such as the number of levels, geometry (cube, hypercube), and the order and number of nested cubes, influence the accuracy of estimations. Employing a generalized inversion method, the author examines models both with and without constraints on parameters, allowing for a comparative analysis of the research outcomes. The interpretation of the properties of these designs is articulated through the use of matrices H and G for unconstrained models, and matrices M(p) and G(p) for constrained models. Specific examples are provided to illustrate the concepts discussed. The findings of the study are presented in graphical form, and a thorough analysis of variance is conducted under the assumption of the admissibility of a linear model without interactions, focusing specifically on cases with fixed levels of factors. This insightful exploration is essential for anyone interested in the intricate relationships between combinatorial structures and statistical analysis.