The purpose of this dissertation is to introduce a version of Stein's method
of exchangeable pairs to solve problems in measure concentration. We
specifically target systems of dependent random variables, since that is where
the power of Stein's method is fully realized. Because the theory is quite
abstract, we have tried to put in as many examples as possible. Some of the
highlighted applications are as follows: (a) We shall find an easily verifiable
condition under which a popular heuristic technique originating from physics,
known as the "mean field equations" method, is valid. No such condition is
currently known. (b) We shall present a way of using couplings to derive
concentration inequalities. Although couplings are routinely used for proving
decay of correlations, no method for using couplings to derive concentration
bounds is available in the literature. This will be used to obtain (c)
concentration inequalities with explicit constants under Dobrushin's condition
of weak dependence. (d) We shall give a method for obtaining concentration of
Haar measures using convergence rates of related random walks on groups. Using
this technique and one of the numerous available results about rates of
convergence of random walks, we will then prove (e) a quantitative version of
Voiculescu's celebrated connection between random matrix theory and free
probability.