This thesis by Daniel Gottesman explores quantum error correction and stabilizer codes, addressing the challenge of controlling errors and decoherence in quantum computation. It provides an overview of quantum error correction, the formalism of stabilizer codes, and discusses known codes, channel capacities, and fault-tolerant quantum computation. The work outlines the structure of quantum codes, their properties, and methods for encoding and decoding. It also covers fault-tolerant computation, concatenated coding, bounds on quantum codes, and examples of stabilizer codes. The thesis emphasizes the importance of quantum error correction in maintaining the integrity of quantum information and discusses the theoretical and practical implications of quantum computing. It also introduces classical coding theory and quantum mechanics as foundational concepts for understanding quantum error correction. The work highlights the role of stabilizer codes in protecting quantum information against errors and the challenges in implementing fault-tolerant quantum computation.This thesis by Daniel Gottesman explores quantum error correction and stabilizer codes, addressing the challenge of controlling errors and decoherence in quantum computation. It provides an overview of quantum error correction, the formalism of stabilizer codes, and discusses known codes, channel capacities, and fault-tolerant quantum computation. The work outlines the structure of quantum codes, their properties, and methods for encoding and decoding. It also covers fault-tolerant computation, concatenated coding, bounds on quantum codes, and examples of stabilizer codes. The thesis emphasizes the importance of quantum error correction in maintaining the integrity of quantum information and discusses the theoretical and practical implications of quantum computing. It also introduces classical coding theory and quantum mechanics as foundational concepts for understanding quantum error correction. The work highlights the role of stabilizer codes in protecting quantum information against errors and the challenges in implementing fault-tolerant quantum computation.