Abstract: This dissertation uses the formalism of category theory to study hybrid phenomena. One begins with a collection of 'non-hybrid' mathematical objects that have been well-studied, together with a notion of how these objects are related to one another; that is, one begins with a category C of the non-hybrid objects of interest. The objects being considered can be 'hybridized' by considering hybrid objects over C consisting of pairs ($D$,A) where $D$ is a small category of a specific form, termed a D-category , which encodes the discrete structure of the hybrid object and A:D→C is a functor encoding its continuous structure. The end result is the category of hybrid objects over C, denoted by Hy(C). In Part I, the foundations for the theory of hybrid objects are established. After reviewing the basics of category theory, inasmuch as they will be needed in this dissertation, D-categories are formally introduced. Hybrid objects over a general category C are then defined along with the corresponding notion of a category of hybrid objects. Elementary properties of categories of this form are discussed. We then proceed to relate the formalism of hybrid objects to hybrid systems in their classical form, the end result of which is a categorical formulation of hybrid systems together with a constructive correspondence between classical hybrid systems and their categorical counterpart. Finally, executions or trajectories of both classical and categorical hybrid systems are introduced, and they are related to one another---again in a constructive fashion. Part II applies the categorical theory of hybrid objects to obtain novel results related to the reduction and stability of hybrid systems. The geometric reduction of simple hybrid systems is first considered, e.g., conditions are given on when robotic systems undergoing impacts can be reduced. As an application of these results, it is shown that a three-dimensional bipedal robotic walker can be reduced to a two-dimensional bipedal walker; the result is walking gaits in three-dimensions based on corresponding walking gaits for a two dimensional biped---walking gaits that simultaneously stabilize the walker to the upright position. Using hybrid objects, the reduction results for simple hybrid systems are generalized to general hybrid systems; to do so, many familiar geometric objects---manifolds, differential forms, et cetera---are first 'hybrizied.' The end result is a hybrid reduction theorem much in the spirt of the classical geometric reduction theorem. This part of the dissertation concludes with a partial characterization of Zeno behavior in hybrid systems. A new type of equilibria, Zeno equilibria, is introduced and sufficient conditions for the stability of these equilibria are given. Since the stability of these equilibria correspond to the existence of Zeno behavior, the end result is sufficient conditions for the existence of Zeno behavior. The final portion of this dissertation, Part III, lays the groundwork for a categorical theory, not of hybrid systems, but of networked systems. It is shown that a network of tagged systems correspond to a network over the category of tagged systems and that taking the composition of such a network is equivalent to taking the limit; this allows us to derive necessary and sufficient conditions for the preservation of semantics, and thus illustrates the possible descriptive power of categories of hybrid and network objects.