The martingale approach after Varadhan and Dolgopyat

We present, in the simplest possible form, the so called martingale problem strategy to establish limit theorems. The presentation is specially adapted to problems arising in partially hyperbolic dynamical systems. We will discuss a simple partially hyperbolic example with fast-slow variables and use the martingale method to prove an averaging theorem and study fluctuations from the average. The emphasis is on ideas rather than on results. Also, no effort whatsoever is done to review the vast literature of the field.