In the past few years we have seen a surge in the theory of finite
Markov chains, by way of new techniques to bounding the convergence
to stationarity. This includes functional techniques such as logarithmic
Sobolev and Nash inequalities, refined spectral and entropy techniques,
and isoperimetric techniques such as the average and blocking conductance and the evolving set methodology. We attempt to give a more or
less self-contained treatment of some of these modern techniques, after
reviewing several preliminaries. We also review classical and modern
lower bounds on mixing times. There have been other important contributions to this theory such as variants on coupling techniques and
decomposition methods, which are not included here; our choice was
to keep the analytical methods as the theme of this presentation. We
illustrate the strength of the main techniques by way of simple examples, a recent result on the Pollard Rho random walk to compute the
discrete logarithm, as well as with an improved analysis of the Thorp
shuffle.