The study of certain classical Markov chains by spectral methods is a classical area going back at least
to work by S.Karlin and J. McGregor. Recently this approach has been undertaken in the case of certain
Quantum Walks. Of particular interest is the relation between the orthogonality measure and the limit law
for the distribution of the displacement of the quantum walker, as well as issues such as recurrence and
localization. We will discuss some general results as well as some specic examples