We discuss sufficient conditions that guarantee the existence of asymptotic expansions for the Central Limit Theorem and Large Deviation Principles for strongly ergodic Markov chains and weakly dependent random variables including observations arising from sufficiently chaotic dynamical systems like piece-wise expanding maps. We primarily use spectral techniques to obtain these results. The work on CLT is joint with Carlangelo Liverani (Rome) and the work on LDPs is joint with Pratima Hebbar (Maryland).