In this seminar I will first set the stage by discussing the phenomenon, sometimes denoted as "condensation of fluctuations", whereby the probability distribution of certain physical quantities develops non-analytical points in the region of rare events. I will show that this is a quite general feature and I will review some simple statistical mechanical models where it is observed. I will discuss how an explanation of the phenomenon can be given in terms of the duality between large deviation events in the given model and typical events in a new and appropriately biased system. Then, I will turn to consider the problem of studying the evolution leading to a large fluctuation. I will do that by introducing and studying analytically a simple model of many identically and independently distributed microscopic variables evolving by means of a master equation. I will show that the process producing a non-typical fluctuation of a variable N is slow and characterized by the power-law growth of the largest possible observable value of N at a given time. I will discuss the analogy between such dynamical process and the slow kinetics observed in systems brought across a phase-transition.