Quantum computation holds the promise of computing problems impossible to solve classically. However, quantum states are very sensitive to environmental noise and operational errors, and therefore the stability of quantum computations is significantly constrained. Here is a general presentation of quantum information protection through error correction and mitigation. We review basic noise models, crucial and sophisticated quantum error correction codes, and useful error reduction methods that are relevant to modern noisy intermediate-scale quantum (NISQ) devices. Quantum chemistry, optimization, and secure communication applications in practice are outlined, with the main toolkits for supporting these techniques. The presentation is finished with forefront trends for realizing scalable, fault-tolerant quantum systems.