This volume presents cutting-edge research capturing foundational and innovative mathematical advances in quantum mechanics. Featuring expert contributions on semiclassical analysis, quantum spin systems, spectral theory, including quasi-periodic operators, quantum information, disordered systems, and other emerging topics, it offers comprehensive insights into modern quantum systems.Compiled from papers presented at the fifteenth QMATH conference held at UC Davis in September 2022, this collection originally appeared in Reviews in Mathematical Physics (Volume 36, Issue 09, October 2024). It is an essential resource for researchers, academics, and graduate students working in mathematical physics, quantum theory, and related interdisciplinary fields.