  In this work we study strong spectral properties of Ruelle transfer operators
related to a large family of Gibbs measures for contact Anosov flows. The
ultimate aim is to establish exponential decay of correlations for H\"older
observables with respect to a very general class of Gibbs measures. The
approach invented in 1997 by Dolgopyat \cite{D1} and further developed in
\cite{St2} is substantially refined here, allowing to deal with much more
general situations than before, although we still restrict ourselves to the
uniformly hyperbolic case. A rather general procedure is established which
produces the desired estimates whenever the Gibbs measure admits a Pesin set
with exponentially small tails, that is a Pesin set whose preimages along the
flow have measures decaying exponentially fast. We call such Gibbs measures
regular. Recent results in \cite{GSt} prove existence of such Pesin sets for
hyperbolic diffeomorphisms and flows for a large variety of Gibbs measures
determined by H\"older continuous potentials. The strong spectral estimates for
Ruelle operators and well-established techniques lead to exponential decay of
correlations for H\"older continuous observables, as well as to some other
consequences such as: (a) existence of a non-zero analytic continuation of the
Ruelle zeta function with a pole at the entropy in a vertical strip containing
the entropy in its interior; (b) a Prime Orbit Theorem with an exponentially
small error.
