Carrier mobility is at the root of our understanding of electronic devices. We present a unified methodology for the parameter-free calculations of phonon-limited drift and Hall carrier mobilities in real materials within the framework of the Boltzmann transport equation. This approach enables accurate and parameter-free calculations of the intrinsic mobility and will find applications in the design of electronic devices under realistic conditions of strain and temperature. The methodology exploits a novel approach for incorporating the effect of long-range quadrupole fields in the electron-phonon scattering rates and capitalizes on a rigorous and efficient procedure for numerical convergence. The accuracy reached in this work allows us to assess the impact of common approximations employed in transport calculations, including the role of exchange and correlation functionals, spin-orbit coupling, pseudopotentials, Wannier interpolation, Brillouin-zone sampling, dipole and quadrupole corrections, and the relaxation-time approximation. We study diamond, silicon, GaAs, 3C-SiC, AlP, GaP, c-BN, AlAs, AlSb, and SrO, and find that our most accurate calculations predict Hall mobilities significantly larger than the experimental data in the case of SiC, AlAs, and GaP. We identify possible improvements to the theoretical and computational frameworks to reduce this discrepancy, and we argue that new experimental data are needed to perform a meaningful comparison, since almost all existing data are more than two decades old. By setting tight standards for reliability and reproducibility, the present work aims to facilitate validation and verification of data and software towards predictive calculations of transport phenomena in semiconductors.