Non-Markovian dynamics in open quantum systems refer to the behavior of quantum systems interacting with environments where memory effects are significant, leading to the revival of quantum properties like coherence and entanglement. Unlike Markovian dynamics, which assume memoryless evolution, non-Markovian processes involve memory effects, where information flows back from the environment to the system. This article discusses the theoretical foundations, mathematical definitions, and physical interpretations of non-Markovian dynamics, emphasizing the quantification of memory effects and their experimental detection. Open quantum systems are described by dynamical maps, which are linear transformations representing the evolution of the system's state. A key concept is divisibility, where a process is Markovian if the dynamical map is P-divisible (positive) or CP-divisible (completely positive). Non-Markovian processes, however, are characterized by non-monotonic behavior of the trace distance between quantum states, indicating information flow back from the environment to the system. The trace distance between two quantum states is a measure of their distinguishability. In non-Markovian dynamics, this distance can increase, reflecting the system's ability to retain and retrieve information from the environment. The article introduces a measure of non-Markovianity, $\mathcal{N}(\Phi)$, which quantifies the maximum total flow of information from the environment back to the system. This measure is derived from the time derivative of the trace distance and is maximized over all possible initial states. The connection between quantum and classical non-Markovianity is explored, highlighting the differences in their definitions and the implications for information flow. The article also discusses the impact of initial system-environment correlations and nonlocal memory effects, emphasizing the importance of these factors in understanding non-Markovian dynamics. Experimental realizations and recent studies on non-Markovian dynamics in photonic and trapped ion systems are reviewed, demonstrating the practical relevance of these concepts. In summary, non-Markovian dynamics in open quantum systems involve complex interactions where memory effects play a crucial role. Theoretical frameworks, such as the trace distance and divisibility of dynamical maps, provide tools to characterize and quantify these effects. Experimental studies further validate these theories, showing the potential for using small open systems as quantum probes to detect and analyze environmental features. The article underscores the significance of non-Markovian dynamics in various quantum applications, from fundamental physics to quantum information and technology.Non-Markovian dynamics in open quantum systems refer to the behavior of quantum systems interacting with environments where memory effects are significant, leading to the revival of quantum properties like coherence and entanglement. Unlike Markovian dynamics, which assume memoryless evolution, non-Markovian processes involve memory effects, where information flows back from the environment to the system. This article discusses the theoretical foundations, mathematical definitions, and physical interpretations of non-Markovian dynamics, emphasizing the quantification of memory effects and their experimental detection. Open quantum systems are described by dynamical maps, which are linear transformations representing the evolution of the system's state. A key concept is divisibility, where a process is Markovian if the dynamical map is P-divisible (positive) or CP-divisible (completely positive). Non-Markovian processes, however, are characterized by non-monotonic behavior of the trace distance between quantum states, indicating information flow back from the environment to the system. The trace distance between two quantum states is a measure of their distinguishability. In non-Markovian dynamics, this distance can increase, reflecting the system's ability to retain and retrieve information from the environment. The article introduces a measure of non-Markovianity, $\mathcal{N}(\Phi)$, which quantifies the maximum total flow of information from the environment back to the system. This measure is derived from the time derivative of the trace distance and is maximized over all possible initial states. The connection between quantum and classical non-Markovianity is explored, highlighting the differences in their definitions and the implications for information flow. The article also discusses the impact of initial system-environment correlations and nonlocal memory effects, emphasizing the importance of these factors in understanding non-Markovian dynamics. Experimental realizations and recent studies on non-Markovian dynamics in photonic and trapped ion systems are reviewed, demonstrating the practical relevance of these concepts. In summary, non-Markovian dynamics in open quantum systems involve complex interactions where memory effects play a crucial role. Theoretical frameworks, such as the trace distance and divisibility of dynamical maps, provide tools to characterize and quantify these effects. Experimental studies further validate these theories, showing the potential for using small open systems as quantum probes to detect and analyze environmental features. The article underscores the significance of non-Markovian dynamics in various quantum applications, from fundamental physics to quantum information and technology.