John Rehr University of Washington There has been dramatic progress in recent years in theories of core-level spectroscopies. Here we discuss the two complementary approaches that build on theories of excited-state electronic structure and response. Both include key many-body effects, which are crucial to a quantitative description. First is a real-space Green's function (RSGF) approach applicable to both core-level x-ray absorption (XAS) and electron energy loss spectra (EELS). This theory is implemented in a relativistic, all-electron code FEFF9 [1], which is applicable to periodic and aperiodic materials alike, throughout the periodic table. The approach can now include ab initio treatments of many-body effects including the screened core-hole interaction, extrinsic and intrinsic inelastic losses, and Debye-Waller factors, optionally with input from auxiliary codes. For example, FEFF9 uses a many-pole GW self-energy (MPSE) to treat final state broadening and self-energy shifts. We also discuss the interpretation of the spectra in terms of geometrical and excited state electronic structure. Second, is a core-level GW/Bethe-Salpeter Equation (GW/BSE) approach, which is implemented in the hybrid code OCEAN [2]. This package use wavefunctions from plane-wave pseudopotential codes (ABINIT or QuantumESPRESSO), the NIST BSE solver, and also the MPSE self-energy. We also discuss recent extensions for treating L-shell spectra and multiplet effects. Though OCEAN is computationally more demanding, recent extensions have greatly improved its efficiency. While OCEAN is limited to near edge spectra, FEFF9 loses accuracy close to an edge; thus the combination permits full-spectrum calculations from soft- to hard-x-ray energies. These approaches are illustrated with a number of applications. References [1] J J Rehr, J J Kas, M P Prange, A P Sorini, Y Takimoto, and F Vila, Comptes Rendus Physique 10, 548 (2009). [2] J Vinson, J J Kas, F D Vila, J J Rehr, and E L Shirley, Phys. Rev. B 85, 045101 (2012). [*]Supported by DOE Grant DE-FG03-97ER45623, and by the DOE CMCSN