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The exploration of metamaterials with artificial sub-wavelength structures has empowered researchers to engineer the propagation of classical waves, enabling advancements in areas such as imaging, sensing, communication, and energy harvesting. Concurrently, the investigation into topology and symmetry has not only unveiled valuable insights into fundamental physics, but also expanded our ability to manipu- late waves effectively. Combined with the remarkable flexibility and diversity of artificial metamaterials, these considerations have sparked a focused research interest. Notably, a class of structures capable of supporting topological propagation modes akin to the Schrödinger equation has been identified. Leveraging metamaterials to emulate Schrödinger dynamics has emerged as a promising avenue for robust wave manipulation and the exploration of quantum phenomena beyond the confines of electronic systems. Despite rapid progress in this burgeoning field, comprehensive summaries are scarce. Thus, this review aims to systematically consolidate recent advancements in classical wave physics based on a Schrödinger equation approach. This discourse initiates with an overview of quantum and classical wave descriptions, subsequently delving into the elucidation of numerous models realized across diverse experimental platforms, including photonic/phononic waveguides, acoustic cavities, and optomechanics. Finally, we address the challenges and prospects associated with emulating Schrödinger dynamics, underscoring the potential for groundbreaking developments in this captivating domain.