An analytic mapping from the unit disc into itself is called inner if it has radial limits of modulus one at almost every point of the unit circle. After the pionnering work of Nevanlinna, the brothers Riesz, Smirnov, Frostman and Beurling, inner functions became a central notion in Complex and Functional Analysis. This minicourse will be focused on dynamical properties of inner functions. We will review the classical Denjoy-Wolff Theorem as well as more modern results on ergodicity and recurrence due to Aaranson, Pommerenke and Doering and Mane. We will introduce the family of Alexandrov-Clark measures of an inner function and discuss some of their main properties. We will use them to show that iterates of an inner function fixing the origin which are not rotations, behave, in a certain sense, as independent random variables. In this direction we will discuss versions of the Central Limit Theorem and of the Khintchine-Kolmogorov pointwise convergence Theorem, for linear combinations of iterates of an inner function.