This chapter presents a review of error correction in quantum computing, a crucial aspect for the development and practical implementation of quantum algorithms. It begins with a brief overview of classical error correction methods, highlighting their applications and the importance of ensuring data integrity in classical systems. Subsequently, the issue of noise in quantum systems is introduced, which significantly affects the fidelity of quantum operations due to the inherent fragility of quantum states. Modern quantum error correction techniques, such as Stabilizer Codes, the Shor Code, and the Steane Code, are explored, detailing how these methods can detect and correct errors in qubits. Specific examples of quantum computing are presented, illustrating the practical implementation of these techniques in quantum algorithms, as well as the current challenges and advancements in noise and error mitigation in quantum computing. This chapter provides a detailed and updated overview of the state-of-the-art in error correction and mitigation within the quantum domain, developed using the PRISMA methodology, emphasizing the importance of these techniques for the future of quantum computing.