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Photon-based spectroscopies have had a significant impact on both fundamentalscience and applications by providing an efficient approach to investigate themicroscopic physics of materials. Together with the development of synchrotronX-ray techniques, theoretical understanding of the spectroscopies themselvesand the underlying physics that they reveal has progressed through advances innumerical methods and scientific computing. In this review, we provide anoverview of theories for angle-resolved photoemission spectroscopy and resonantinelastic X-ray scattering applied to quantum materials. First, we discussmethods for studying equilibrium spectroscopies, including first-principlesapproaches, numerical many-body methods and a few analytical advances. Second,we assess the recent development of ultrafast techniques for out-of-equilibriumspectroscopies, from characterizing equilibrium properties to generatingtransient or metastable states, mainly from a theoretical point of view.Finally, we identify the main challenges and provide an outlook for the futuredirection of the field.

Theoretical Understanding of Photon Spectroscopies in Correlated Materials In and Out of Equilibrium