  We study Coulomb drag between the top and bottom surfaces of topological
insulator films. We derive a kinetic equation for the thin-film spin density
matrix containing the full spin structure of the two-layer system, and analyze
the electron-electron interaction in detail in order to recover all terms
responsible for Coulomb drag. Focusing on typical topological insulator
systems, with film thicknesses d up to 6 nm, we obtain numerical and
approximate analytical results for the drag resistivity $\rho_\text{D}$ and
find that $\rho_\text{D}$ is proportional to
$T^2d^{-4}n^{-3/2}_{\text{a}}n^{-3/2}_{\text{p}}$ at low temperature T and low
electron density $n_{\text{a,p}}$, with a denoting the active layer and p the
passive layer. In addition, we compare $\rho_{\text{D}}$ with graphene,
identifying qualitative and quantitative differences, and we discuss the multi
valley case, ultra thin films and electron-hole layers.
