Optimizing cost functions under dynamic constraints has been widely studied for over 70 years, with applications across engineering, medicine, and biology. A key challenge arises when adversarial agents, designed to oppose the main control objective, are involved. This scenario is often modeled using differential games, where constraints are governed by differential equations. Dynamic constraints in modern applications that combine physics, computing, and networks often exhibit both continuous and discrete behavior, influenced by nonsmooth factors like intermittent information and resets of variables. These constraints are well-suited to hybrid system models, which combine continuous and discrete dynamics. However, designing algorithms that ensure optimality under these hybrid constraints requires new methods, as existing tools from differential games may lead to suboptimal solutions. This dissertation aims to address the lack of tools for designing algorithms for hybrid games with dynamic constraints, specifically beyond those modeled by finite-state automata or switched systems. First, we formulate a framework for the study of two-player zero-sum games under dynamic constraints given in terms of hybrid dynamical systems. We employ our framework to study games with different types of termination conditions. Analyzing the case in which solutions to the hybrid system are complete allows us to propose results on optimality and asymptotic stability for games over the infinite horizon with applications to security and disturbance rejection problems. By considering the more general case of games over a finite horizon, we employ existing tools in hybrid systems to design optimal strategies upon appropriate specifications of terminal hybrid time and terminal state sets. We study input-to-state stability and safety of hybrid systems under disturbances as inverse-optimal two-player zero-sum games. We propose QP-based controls in terms of Lyapunov and barrier functions to construct a meaningful cost functional that is minimized under the worst-case disturbance. For multi-stage hybrid games, an optimality analysis is proposed with applications to capture-the-flag games. Finally, imperfect state information motivates the study of optimal designs of control strategies together with state observers.