This book opens new ways for probing electronic correlations, emergent phenomena, and unconventional metallic states. The Fermi surface governs the fundamental properties of metals. Quantum oscillation measurements are the most precise method for determining Fermi surface properties in bulk three-dimensional metals, owing to Onsager's relation, which connects the oscillation frequency to an extremal cross-section of the Fermi surface. This canonical, semiclassical theory of quantum oscillations has been remarkably successful in unearthing electronic properties of metals for over 75 years. However, with the emergence of modern quantum materials and advancements in experimental sensitivity, breakdowns of the semiclassical theory have recently come to light. In this thesis, we demonstrate that additional quantum oscillation frequencies--breaking Onsager's relation--emerge generically in multiband metals. These anomalous frequencies, referred to as non-Onsager oscillations, are induced by impurities, scattering from fluctuations, or weak and strong electronic interactions. Using a combination of analytic theory, numeric modeling, and experimental verification, we establish them as fundamental features of quantum oscillation spectra.

Properly identifying non-Onsager oscillations is essential for accurately reconstructing the Fermi surface. Moreover, these frequencies encode rich information about underlying material properties beyond the scope of the semiclassical theory. The results of this book refine our understanding of quantum oscillations in complex materials and open new avenues for probing electronic correlations, emergent phenomena, and unconventional metallic states.