In this dissertation, we study stable and strongly stable matching for the Cyclic Matching Problem when the number of dimensions (sides) is greater than or equal to three. The following is a summary of the analysis and results.; 1. We offer preliminary results on Pareto optimality in higher dimensions; we also comment on changes in the relationships among stability, strong stability, and Pareto optimality, when we move from the two-sided to the higher-sided case.; 2. We construct a function whose fixed points are exactly the set of strongly stable matchings.; 3. We show how to compute all strongly stable matchings for a given instance of the Cyclic Matching Problem using the methodology of Echenique and Yenmez (2007).; 4. We construct a non-monotonic function whose fixed points are exactly the set of stable matchings.; 5. We investigate an extension of the Adachi program to higher dimensions; in particular, we look at the properties of the fixed points of certain functions constructed so that they are increasing with respect to "natural" partial orders in higher dimensions.; 6. We consider extensions of the Gale-Shapley algorithm to higher dimensions; we design the Hierarchical Gale-Shapley Algorithm and comment on its properties with regard to (1) convergence and (2) stability of the output matching.