The article discusses the dynamics of open quantum systems, focusing on non-Markovian effects and their applications. It begins by explaining the standard treatment of open quantum systems using a dynamical semigroup and a master equation in Lindblad form, which describes memoryless dynamics leading to irreversible loss of quantum features. However, many applications require a more sophisticated description due to the presence of nontrivial memory effects and nonlocal quantum properties such as quantum coherence, correlations, and entanglement. The authors introduce the concept of quantum non-Markovianity, which is distinct from classical Markovianity and cannot be directly transferred to the quantum regime. They define quantum non-Markovianity rigorously and discuss its physical interpretation and classification. The article also explores various models and applications of non-Markovianity, including pure decoherence dynamics, dissipative processes, and the spin-boson model. It highlights how memory effects in open system dynamics reflect characteristic properties of the environment, suggesting that small open systems can be used as quantum probes to detect nontrivial features of complex environments. The article further examines the impact of initial system-environment correlations and nonlocal memory effects, and provides experimental realizations of non-Markovian quantum dynamics. It discusses recent experiments that demonstrate the transition from Markovian to non-Markovian dynamics, the influence of nonlocal environmental correlations, and local detection schemes for nonclassical initial system-environment correlations. Finally, the article reviews the mathematical definitions and measures of quantum non-Markovianity, including the trace distance and the generalized measure based on the Helstrom matrix. It establishes connections between quantum and classical non-Markovianity and discusses the relationship between quantum Markovianity and the divisibility of the dynamical map.The article discusses the dynamics of open quantum systems, focusing on non-Markovian effects and their applications. It begins by explaining the standard treatment of open quantum systems using a dynamical semigroup and a master equation in Lindblad form, which describes memoryless dynamics leading to irreversible loss of quantum features. However, many applications require a more sophisticated description due to the presence of nontrivial memory effects and nonlocal quantum properties such as quantum coherence, correlations, and entanglement. The authors introduce the concept of quantum non-Markovianity, which is distinct from classical Markovianity and cannot be directly transferred to the quantum regime. They define quantum non-Markovianity rigorously and discuss its physical interpretation and classification. The article also explores various models and applications of non-Markovianity, including pure decoherence dynamics, dissipative processes, and the spin-boson model. It highlights how memory effects in open system dynamics reflect characteristic properties of the environment, suggesting that small open systems can be used as quantum probes to detect nontrivial features of complex environments. The article further examines the impact of initial system-environment correlations and nonlocal memory effects, and provides experimental realizations of non-Markovian quantum dynamics. It discusses recent experiments that demonstrate the transition from Markovian to non-Markovian dynamics, the influence of nonlocal environmental correlations, and local detection schemes for nonclassical initial system-environment correlations. Finally, the article reviews the mathematical definitions and measures of quantum non-Markovianity, including the trace distance and the generalized measure based on the Helstrom matrix. It establishes connections between quantum and classical non-Markovianity and discusses the relationship between quantum Markovianity and the divisibility of the dynamical map.