Quantum error correction for quantum memories is a critical component in building a universal quantum computer. This review discusses the formalism of qubit stabilizer and subsystem stabilizer codes and their application in protecting quantum information. It covers the theory of fault tolerance and quantum error correction, examples of various codes, the general quantum error correction conditions, the noise threshold, the role of Clifford gates, and the path to fault-tolerant quantum computation. The second part of the review focuses on two-dimensional (topological) codes, particularly the surface code architecture, discussing decoding complexity, passive or self-correcting quantum memories, and the route to fault-tolerant universal quantum computation. The review begins by introducing the concept of quantum error correction, starting with Shor's code as the most basic example of a quantum error correction code. It then discusses the formalism of stabilizer codes, including subsystem stabilizer codes, and the general quantum error correction conditions. The review also covers the physical error models, the amplitude damping code, and the qubit-into-oscillator code. It discusses the D-dimensional (stabilizer) codes and the relationship between error correction and fault tolerance. The review then focuses on 2D (topological) error correction, particularly the surface code. It discusses the surface code's ability to correct errors, the preparation and measurement of logical qubits, and the implementation of logical gates such as the Hadamard gate and the CNOT gate. It also covers different 2D code constructions, including the Bacon-Shor code, the surface code with harmonic oscillators, and the subsystem surface code. The review discusses decoding and parity check measurements, the physical locality of decoding, and the concept of passive or self-correction. The review concludes with a discussion of future challenges in quantum error correction and recommends a reference book on the topic. It also discusses error mitigation techniques, including passive error mitigation and encoding quantum information in many-body quantum systems. The review highlights the importance of quantum error correction in achieving fault-tolerant quantum computation and the role of various codes in this endeavor.Quantum error correction for quantum memories is a critical component in building a universal quantum computer. This review discusses the formalism of qubit stabilizer and subsystem stabilizer codes and their application in protecting quantum information. It covers the theory of fault tolerance and quantum error correction, examples of various codes, the general quantum error correction conditions, the noise threshold, the role of Clifford gates, and the path to fault-tolerant quantum computation. The second part of the review focuses on two-dimensional (topological) codes, particularly the surface code architecture, discussing decoding complexity, passive or self-correcting quantum memories, and the route to fault-tolerant universal quantum computation. The review begins by introducing the concept of quantum error correction, starting with Shor's code as the most basic example of a quantum error correction code. It then discusses the formalism of stabilizer codes, including subsystem stabilizer codes, and the general quantum error correction conditions. The review also covers the physical error models, the amplitude damping code, and the qubit-into-oscillator code. It discusses the D-dimensional (stabilizer) codes and the relationship between error correction and fault tolerance. The review then focuses on 2D (topological) error correction, particularly the surface code. It discusses the surface code's ability to correct errors, the preparation and measurement of logical qubits, and the implementation of logical gates such as the Hadamard gate and the CNOT gate. It also covers different 2D code constructions, including the Bacon-Shor code, the surface code with harmonic oscillators, and the subsystem surface code. The review discusses decoding and parity check measurements, the physical locality of decoding, and the concept of passive or self-correction. The review concludes with a discussion of future challenges in quantum error correction and recommends a reference book on the topic. It also discusses error mitigation techniques, including passive error mitigation and encoding quantum information in many-body quantum systems. The review highlights the importance of quantum error correction in achieving fault-tolerant quantum computation and the role of various codes in this endeavor.