<p>Flexural waves constitute a fundamental class of elastic waves in thin structures and play a central role in vibration, noise, and energy transport across a wide range of engineering systems. This review provides a comprehensive overview of the theoretical foundations and manipulation strategies of flexural waves. We begin with the classical Kirchhoff–Love plate theory and related models, dispersion characteristics, and range of validity. Building on this foundation, we summarize recent advances in flexural-wave manipulation using elastic metasurfaces, which enable anomalous reflection, wavefront shaping, and mode conversion through subwavelength structural design. We further highlight emerging developments in topological physics and bound states in the continuum (BICs) within flexural wave systems, including defect-immune transport governed by lattice topology, non-radiating embedded modes, and high-quality (high-<i>Q</i>) factor resonances. These approaches illustrate the rapid evolution of flexural waves from classical vibration theory to a modern multidisciplinary field. The review concludes by outlining key challenges and promising research directions, including integrated wave-control strategies and emerging concepts for next-generation flexural-wave devices.</p>