  Photon-based spectroscopies have had a significant impact on both fundamental
science and applications by providing an efficient approach to investigate the
microscopic physics of materials. Together with the development of synchrotron
X-ray techniques, theoretical understanding of the spectroscopies themselves
and the underlying physics that they reveal has progressed through advances in
numerical methods and scientific computing. In this review, we provide an
overview of theories for angle-resolved photoemission spectroscopy and resonant
inelastic X-ray scattering applied to quantum materials. First, we discuss
methods for studying equilibrium spectroscopies, including first-principles
approaches, numerical many-body methods and a few analytical advances. Second,
we assess the recent development of ultrafast techniques for out-of-equilibrium
spectroscopies, from characterizing equilibrium properties to generating
transient or metastable states, mainly from a theoretical point of view.
Finally, we identify the main challenges and provide an outlook for the future
direction of the field.
