This dissertation is the collection of a progressive research on the topic of topological quantum computation and information with the focus on the error threshold of the well-known models such as the unpaired Majorana, the toric code, and the planar code. We study the basics of quantum computation and quantum information, and in particular quantum error correction. Quantum error correction provides a tool for enhancing the quantum computation fidelity in the noisy environment of a real world. We begin with a brief introduction to stabilizer codes. The stabilizer formalism of the theory of quantum error correction gives a well-defined description of quantum codes that is used throughout this dissertation. Then, we turn our attention to a quite new subject, namely, topological quantum codes. Topological quantum codes take advantage of the topological characteristics of a physical many-body system. The physical many-body systems studied in the context of topological quantum codes are of two essential natures: they either have intrinsic interaction that self-corrects errors, or are actively corrected to be maintained in a desired quantum state. Examples of the former are the toric code and the unpaired Majorana, while an example for the latter is the surface code. A brief introduction and history of topological phenomena in condensed matter is provided. The unpaired Majorana and the Kitaev toy model are briefly explained. Later we introduce a spin model that maps onto the Kitaev toy model through a sequence of transformations. We show how this model is robust and tolerates local perturbations. The research on this topic, at the time of writing this dissertation, is still incomplete and only preliminary results are represented. As another example of passive error correcting codes with intrinsic Hamiltonian, the toric code is introduced. We also analyze the dynamics of the errors in the toric code known as anyons. We show numerically how the addition of disorder to the physical system underlying the toric code slows down the dynamics of the anyons. We go further and numerically analyze the presence of time-dependent noise and the consequent delocalization of localized errors. The main portion of this dissertation is dedicated to the surface code. We study the surface code coupled to a non-interacting bosonic bath. We show how the interaction between the code and the bosonic bath can effectively induce correlated errors. These correlated errors may be corrected up to some extend. The extension beyond which quantum error correction seems impossible is the error threshold of the code. This threshold is analyzed by mapping the effective correlated error model onto a statistical model. We then study the phase transition in the statistical model. The analysis is in two parts. First, we carry out derivation of the effective correlated model, its mapping onto a statistical model, and perform an exact numerical analysis. Second, we employ a Monte Carlo method to extend the numerical analysis to large system size. We also tackle the problem of surface code with correlated and single-qubit errors by an exact mapping onto a two-dimensional Ising model with boundary fields. We show how the phase transition point in one model, the Ising model, coincides with the intrinsic error threshold of the other model, the surface code.