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In this review we present the theoretical foundations and first principlesframeworks to describe quantum matter within quantum electrodynamics (QED) inthe low-energy regime. Having a rigorous and fully quantized description ofinteracting photons, electrons and nuclei/ions, from weak to stronglight-matter coupling regimes, is pivotal for a detailed understanding of theemerging fields of polaritonic chemistry and cavity materials engineering. Theuse of rigorous first principles avoids ambiguities and problems stemming fromusing approximate models based on phenomenological descriptions of light,matter and their interactions. By starting from fundamental physical andmathematical principles, we first review in great detail non-relativistic QED,which allows to study polaritonic systems non-perturbatively by solving aSchr\"odinger-type equation. The resulting Pauli-Fierz quantum field theoryserves as a cornerstone for the development of computational methods, such asquantum-electrodynamical density functional theory, QED coupled cluster orcavity Born-Oppenheimer molecular dynamics. These methods treat light andmatter on equal footing and have the same level of accuracy and reliability asestablished methods of computational chemistry and electronic structure theory.After an overview of the key-ideas behind those novel ab initio QED methods, weexplain their benefits for a better understanding of photon-induced changes ofchemical properties and reactions. Based on results obtained by ab initio QEDmethods we identify the open theoretical questions and how a so far missingmechanistic understanding of polaritonic chemistry can be established. Wefinally give an outlook on future directions within polaritonic chemistry andfirst principles QED and address the open questions that need to be solved inthe next years both from a theoretical as well as experimental viewpoint.

Understanding polaritonic chemistry from ab initio quantum electrodynamics