Quantum computation is a proposed model of computation that applies quantummechanics to perform information processing and store information in quantumstates. Quantum mechanics applies for many different phenomena, withmany possible systems in which it is possible to model and manipulate thefundamental quantum information bit - the qubit - and thus there are hypotheticallymany ways to construct a quantum computer. One proposed way ofquantum computation is to use non-abelian anyons to model qubits. These areexotic quasi-particles whose wave functions evolve non-trivially when permutingtheir positions. This allows for computation with qubits simply by permutinganyons, a process called braiding since their trajectories in spacetime resemblebraids. The quantum states associated with the anyons evolve only when thepositions of anyons are permuted and do not depend on the paths the anyonstake. For this reason this model of quantum computation is called topologicalquantum computation (TQC). One of the main advantages of TQC is that computationsare inherently fault tolerant: there is no noise due to anyons takingstrange paths since the quantum evolution is path independent. The goal ofthis text is to investigate the mathematical framework for this proposed modelof quantum computation. The main results are the possible gates that can beapplied to two one-qubit topological computers. Given the key properties ofthe anyons used, Theorem 5.2 states the possible one-qubit gates in an Isingcomputer, and Theorem 7.1 states the possible one-qubit gates in a Fibonaccicomputer.