  This article reviews recent theoretical and experimental work on transport
due to the surface states of three-dimensional topological insulators. The
theoretical focus is on longitudinal transport in the presence of an electric
field, including Boltzmann transport, quantum corrections and weak
localization, as well as longitudinal and Hall transport in the presence of
both electric and magnetic fields and/or magnetizations. Special attention is
paid to transport at finite doping, to the $\pi$-Berry phase, which leads to
the absence of backscattering, Klein tunneling and half-quantized Hall
response. Signatures of surface states in ordinary transport and
magnetotransport are clearly identified. The review also covers transport
experiments of the past years, reviewing the initial obscuring of surface
transport by bulk transport, and the way transport due to the surface states
has increasingly been identified experimentally. Current and likely future
experimental challenges are given prominence and the current status of the
field is assessed.
