  We construct a theory of charge transport by the surface states of
topological insulators in three dimensions. The focus is on the experimentally
relevant case when the electron doping is such that the Fermi energy
$\epsilon_F$ and transport scattering time $\tau$ satisfy $\epsilon_F
\tau/\hbar \gg 1$, but sufficiently low that $\epsilon_F$ lies below the bottom
of the conduction band. Our theory is based on the spin density matrix and
takes the quantum Liouville equation as its starting point. The scattering term
is determined accurately to linear order in the impurity density. We consider
scattering by charged impurities and short-range scatterers such as surface
roughness. We calculate also the polarization function in topological
insulators, emphasizing the differences from graphene. We find that the main
contribution to the conductivity is $\propto n_i^{-1}$, where $n_i$ is the
impurity density, and will have different carrier density dependencies for
different forms of scattering. Two different contributions to this conductivity
are traced to the scalar and spin-dependent terms in the Hamiltonian and their
relative weight depends on the doping density. Our results contain all
contributions to the conductivity to orders zero and one in the impurity
density. We discuss also a way to determine the dominant scattering angles by
studying the ratio of the transport relaxation time to the Bloch lifetime as a
function of the Wigner-Seitz radius $r_s$. We also discuss the effect on the
surface states of adding metallic contacts.
