We briefly review some aspects of the anomalous diffusion, and its relevance in reactive systems. In particular we consider strong anomalous diffusion characterized by the moment behaviour <(x(t)(q)> similar to t(qv)(q), where v(q) is a non constant function, and we discuss its consequences. Even in the apparently simple case v(2) = 1/2, strong anomalous diffusion may correspond to non trivial features, such as non Gaussian probability distribution and peculiar scaling of large order moments.

This book reviews the basic ideas of the Law of Large Numbers with its consequences to the deterministic world and the issue of ergodicity. Applications of Large Deviations and their outcomes to Physics are surveyed. The book covers topics encompassing ergodicity and its breaking and the modern applications of Large deviations to equilibrium and non-equilibrium statistical physics, disordered and chaotic systems, and turbulence.

This contribution aims at introducing the topics of this book. We start with a brief historical excursion on the developments from the law of large numbers to the central limit theorem and large deviations theory. The same topics are then presented using the language of probability theory. Finally, some applications of large deviations theory in physics are briefly discussed through examples taken from statistical mechanics, dynamical and disordered systems.