Quantum spin systems have been providing modern physics with ever new challenges to tackle in the past several decades; from the theoretical postulation of topological quantum order, a new chapter was revealed in the understanding of phase transitions in quantum systems. The novel properties of topologically ordered systems had caught the interest of quantum information scientists, and research into their potential in the field of quantum computing soon began. Nowadays, this utility of topologically ordered systems is thoroughly researched in the field of topological quantum computing. Systems that exhibit topological order present a valuable resource for quantum computing – by use of the global observables that remain robust when such a system is acted on by a local perturbation. Under these conditions, the information stored in the system remains stable, providing a unique modus of circumventing the problems that plague standard quantum computing schemes. To improve on the applicability of such systems in fault-tolerant quantum computation, there exists a need to understand the exact behavior of the encoded information under perturbation. To this end, this thesis presents critical mathematical methods used to research the dynamics of local quantum systems — the quasi-adiabatic continuation, the Lieb-Robinson bounds, and the self-energy expansion—and utilizes them to further the understanding of the effect of weak local perturbations on the information stored in a topologically ordered system. In particular, the influence of perturbation on topologically ordered systems with local puncture defects is examined, and general bounds on the stability of topological order in such systems are derived.