  This paper proposes a quantitative description of the low energy edge states
at the interface between two-dimensional topological insulators. They are
modeled by continuous Hamiltonians as systems of Dirac equations that are
amenable to a large class of random perturbations. We consider general as well
as fermionic time reversal symmetric models. In the former case, Hamiltonians
are classified using the index of a Fredholm operator. In the latter case, the
classification involves a ${\mathbb Z}_2$ index. These indices dictate the
number of topologically protected edge states.
  A remarkable feature of topological insulators is the asymmetry (chirality)
of the edge states, with more modes propagating, say, up than down. In some
cases, backscattering off imperfections is prevented when no mode can carry
signals backwards. This is a desirable feature from an engineering perspective,
which raises the question of how back-scattering is protected topologically. A
major motivation for the derivation of continuous models is to answer such a
question.
  We quantify how backscattering is affected but not suppressed by the
non-trivial topology by introducing a scattering problem along the edge and
describing the effects of topology and randomness on the scattering matrix.
Explicit macroscopic models are then obtained within the diffusion
approximation of field propagation to show the following: the combination of
topology and randomness results in un-hindered transport of the protected modes
while all other modes (Anderson) localize.
