Koopman operator theory offers a rigorous treatment of dynamics and has been emerging as an alternative modeling and learning-based control method across various robotics sub-domains. Due to its ability to represent nonlinear dynamics as a linear (but higher-dimensional) operator, Koopman theory offers a fresh lens through which to understand and tackle the modeling and control of complex robotic systems. Moreover, it enables incremental updates and is computationally inexpensive, thus making it particularly appealing for real-time applications and online active learning. This review delves deeply into the foundations of Koopman operator theory and systematically builds a bridge from theoretical principles to practical robotic applications. We begin by explaining the mathematical underpinnings of the Koopman framework and discussing approximation approaches for incorporating inputs into Koopman-based modeling. Foundational considerations, such as data collection strategies as well as the design of lifting functions for effective system embedding, are also discussed. We then explore how Koopman-based models serve as a unifying tool for a range of robotics tasks, including model-based control, real-time state estimation, and motion planning. The review proceeds to a survey of cutting-edge research that demonstrates the versatility and growing impact of Koopman methods across diverse robotics sub-domains: from aerial and legged platforms to manipulators, soft-bodied systems, and multi-agent networks. A presentation of more advanced theoretical topics, necessary to push forward the overall framework, is included. Finally, we reflect on some key open challenges that remain and articulate future research directions that will shape the next phase of Koopman-inspired robotics. To support practical adoption, we provide a hands-on tutorial with executable code at https://shorturl.at/ouE59.