  One of the fundamental properties of semiconductors is their ability to
support highly tunable electric currents in the presence of electric fields or
carrier concentration gradients. These properties are described by transport
coefficients such as electron and hole mobilities. Recently, advances in
electronic structure methods for real materials have made it possible to study
these properties with predictive accuracy and without resorting to empirical
parameters. Here, we review the most recent developments in the area of ab
initio calculations of carrier mobilities of semiconductors. In the first part,
we offer a brief historical overview of approaches to the calculation of
carrier mobilities, and we establish the conceptual framework underlying modern
ab initio approaches. We summarize the Boltzmann theory of carrier transport
and we discuss its scope of applicability, merits, and limitations in the
broader context of many-body Green's function approaches. We discuss recent
implementations of the Boltzmann formalism within the context of density
functional theory and many-body perturbation theory calculations, placing an
emphasis on the key computational challenges and suggested solutions. In the
second part, we discuss recent investigations of classic materials such as
silicon, diamond, GaAs, GaN, Ga2O3, and lead halide perovskites as well as
low-dimensional semiconductors such as graphene, silicene, phosphorene, MoS2,
and InSe. We also review recent efforts toward high-throughput calculations of
carrier transport. In the last part, we discuss the extension of the
methodology to study spintronics and topological materials and we comment on
the possibility of incorporating Berry-phase effects and many-body correlations
beyond the standard Boltzmann formalism.
