The physical relevance of fluctuations in a probabilistic language demands the illustration of basic mathematical tools, including the central limit theorem and the theory of large deviations. A short summary about random matrix theory precedes the model of generalized random walks, which includes Levy flights and walks as representations of anomalous diffusive processes. Einstein's approach to the role of fluctuations in thermodynamic processes is detailed for both an isolated and a thermalized thermodynamic system. An introduction to stochastic thermodynamics and to generalized fluctuation theorems is finally discussed.

An interface is rough if the mean square fluctuations of its position diverge at large times and system sizes. This may occur when the interface is driven out of equilibrium in the presence of some noise and the way roughness diverges defines suitable critical exponents. We introduce and discuss extensively two important universality classes: the Edwards–Wilkinson and the Kardar–Parisi–Zhang. The latter has been the subject of renewed interest since it was possible to determine analytically the whole spectrum of fluctuations and it was found an experimental system satisfying such predictions with great accuracy. The last part of the chapter is devoted to nonlocal models, specifically the celebrated Diffusion Limited Aggregation.

The phenomenological theory proposed by Einstein for interpreting the phenomenon of Brownian motion is described in detail. The alternative approaches due to Langevin and Fokker–Planck are also illustrated. The theory of Markov chains is also reported as a basic mathematical approach to stochastic processes in discrete space and time; various of its applications, for example, the Monte Carlo method, are also illustrated. The theory of stochastic equations, as a representation of stochastic processes in continuous space–time, is discussed and used for obtaining a generalized, rigorous formulation of the Langevin and Fokker–Planck equations for generalized fluctuating observables. The Arrhenius formula as an example of the first exit-time problem is also derived.

This final chapter is a short introduction to pattern-forming systems, which highlights a few concepts and models rather than pretending to give a general overview (which is impossible in 40 pages). We focus on stationary bifurcations, distinguishing between scenarios where the critical wavevector vanishes and where it is a finite value, because they have different nonlinear behaviors. A few pages are devoted to describe some different experimental setups: thermal convection (a fluid heated from below, showing the rising of convection cells); unstable growth process (under particle deposition, with the formation of mounds); and a rotating mixture of granular systems (with their phase separation).

This chapter essentially faces the following question: If at equilibrium a system has a phase transition between a disordered phase and an ordered phase, how does it relax to equilibrium if it is quenched from the former to the latter phase? Quenching means that an external parameter, typically the temperature, is suddenly changed. The answer depends on some relevant factors: if dynamics conserves or not the order parameter; if the order parameter is a scalar or a vector; if long-range interactions are present or not. We devote special attention to the short-range Ising model, but we also consider nonscalar systems. If the order parameter is conserved, its value before quenching is also an important parameter, allowing to distinguish between two different trigger mechanisms of the relaxation process: spinodal decomposition and thermally activated nucleation.