Poincaré's geometric representation has long been fundamental in dynamical system analysis. However, its limitations in handling high-dimensional and uncertain systems have become increasingly apparent in modern engineering and data analysis. This article comprehensively reviews Koopman Operator Theory (KOT) and Dynamic Mode Decomposition (DMD) in the context of data-driven science and engineering. The survey introduces a conceptual shift towards the dynamics of observables, focusing on the Koopman operator's ability to capture nonlinear dynamics in infinite-dimensional space. Examining spectral properties highlights the interconnectedness of various methodologies, aiming to enhance accessibility and applicability. The potential transition of KOT from theoretical to practical applications is underscored, emphasizing its efficacy in numerical analysis and industrial contexts. Drawing from the geometric framework established by Poincaré, the limitations of traditional approaches in handling high-dimensional and uncertain systems are discussed. Ergodic and operator theories offer elegant solutions to these challenges, enabling linear representations of nonlinear dynamics without sacrificing information. The practical considerations and computational challenges inherent in utilizing Koopman and Perron-Frobenius operators are explored, emphasizing the trade-offs between accuracy and computational efficiency. The article also addresses the rising importance of data-driven methodologies, particularly in the era of big data and machine learning. The Koopman operator theory is a promising approach for unsupervised learning in dynamically evolving systems, offering insights into system behaviour from limited data. This review provides a historical overview, theoretical foundation, and practical implications of Koopman operator theory and dynamic mode decomposition. It positions them as powerful tools for data-driven analysis and engineering design in complex dynamical systems.