This is a rich chapter in which we delve into the study of the (weak and strong) laws of large numbers, and of the central limit theorem. The latter is first considered for sums of independent stochastic variables whose distributions have a finite variance, and then for variables with diverging variance. Several appendices report on both basic mathematical tools and lengthy details of computation. Among the first, the rules of variable change in probability are presented, Fourier and Laplace transforms are introduced, and their role as generating functionals of moments and cumulants, and the different kinds of convergence of stochastic functions are considered and exemplified.