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.