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Self-reconfigurable robotic systems are variable-morphology machines capable of changing their overall structure by rearranging the modules they are composed of. Individual modules are capable of connecting and disconnecting to and from one another, which allows the robot to adapt to changing environments. Optimally reconfiguring such systems is computationally prohibitive and thus in general self-reconfiguration approaches aim at approximating optimal solutions. Nonetheless, even for approximate solutions, centralized methods scale poorly in the number of modules. Therefore, the objective of this research is the development of decentralized self-reconfiguration methods for modular robotic systems. Building on completeness results of the centralized algorithms in this work, decentralized methods are developed that guarantee stochastic convergence to a given target shape. A game-theoretic approach lays the theoretical foundation of a novel potential game-based formulation of the self-reconfiguration problem. Furthermore, two extensions to the basic game-theoretic algorithm are proposed that enable agents to modify the algorithms' parameters during runtime and improve convergence times. The flexibility in the choice of utility functions together with runtime adaptability makes the presented approach and the underlying theory suitable for a range of problems that rely on decentralized local control to guarantee global, emerging properties. The experimental evaluation of the presented algorithms relies on a newly developed multi-robotic testbed called the "Robotarium" that is equipped with custom-designed miniature robots, the "GRITSBots". The Robotarium provides hardware validation of self-reconfiguration on robots but more importantly introduces a novel paradigm for remote accessibility of multi-agent testbeds with the goal of lowering the barrier to entrance into the field of multi-robot research and education.