We present a formalism for the resonant inelastic X-ray scattering (RIXS) cross section. The resulting compact expression in terms of polarizability matrix elements, particularly lends itself to the implementation in an all-electron many-body perturbation theory (MBPT) framework, which is realized in the full-potential package exciting. With the carbon K edge RIXS of diamond and the oxygen K edge RIXS of β-Ga2O3, respectively, we demonstrate the importance of electron-hole correlation and atomic coherence in the RIXS spectra.