This thesis investigates various topological phases of matter in two-dimensional and quasi one-dimensional systems. These exotic states of matter have applications in topological quantum computation; which is an inherently fault-tolerant quantum computation scheme. In these schemes, computations are implemented by braiding anyonic excitations. In this thesis we examine three aspects of braiding: braiding of anyonic excitations on graphs, topological lattice models, and non-adiabatic perturbations of a qubit constructed from Majorana bound states. In the first part of the thesis we introduce a universal framework to discuss the braiding of anyonic excitations on graphs as a model of a quantum wire network. We show that many features of the planar algebraic theory of anyons may be extended to graphs. In this direction, we introduce graph hexagon equations, a generalisation of the planar hexagon equations and demonstrate that this framework has several similarities and differences from its planar counterpart. Notably, depending on the graph, we find solutions that do not exist in the planar theory. We study this framework on a variety of graphs and tabulate solutions. In the second part, we investigated non-adiabatic perturbations of a topological memory that consists of two p-wave superconducting wires separated by a nontopological junction. We consider noise in the potential creating the non-trivial topological phase and also the effect of shuttling the Majoranas, a necessary step in braiding. We examine a mechanism for bit and phase flip errors where excitations from one wire tunnel through a junction into another wire, we also outline a scheme that utilises disorder to minimise such situations. In the final part of the thesis, we construct a modified toric code from Hopf algebra gauge theory. We find that introducing a non-trivial quasitriangular structure on the gauge group changes the identification of braiding statistics in the quantum double, although it is of the same topological order as the toric code. In particular, when the gauge group is CZN we can interpret this as a form of flux attachment, where under exchange, the electric charges behave as if they have fluxes attached.