"Research in numerous scientific fields, including many of the fundamental research frontiers in science and engineering, has produced large empirical data sets with highly complex structure. There, the search for efficient ways of detecting and evaluating relevant information is currently one of the dominant problems, and the use of sophisticated statistical methods has been advanced as a necessary and central part of the analysis. In particular, recent advances in Markov chain Monte Carlo (MCMC) methods have revolutionized statistical analysis, vastly increasing its impact. But in certain important classes of applications, MCMC algorithms fail. The core aim of the proposed work is to battle the main obstacle for their successful application, namely their slow convergence rate. We propose the development of a new theoretical and methodological approach for MCMC variance reduction, based on control variates. This has been considered an important but very difficult area, and it remains virtually virgin territory. Nevertheless, we have exciting preliminary results clearly showing that, in certain cases, the variance of common MCMC algorithms is reduced by a factor of as much as a million. The fundamental objectives of the proposed research are: 1) To develop the necessary mathematical foundation for the application of control variates to Markov chains; 2) Introduce generic methodologies for the direct use of control variates in all the major families of statistical MCMC algorithms currently used in applications; 3) Apply these new methodologies to make significant contributions across the range of scientific areas that require intensive use of statistical computing via MCMC, including genetics, finance, engineering, and medicine. In view of the enormous practical importance of the applications considered, the potential impact of even moderate theoretical or methodological advances can hardly be overestimated."