This is a text about probabilistic approximations, which are mathematical statements providing estimates of the distance between the laws of two random objects. As the title suggests, we will be mainly interested in approximations involving one or more normal (equivalently called Gaussian) random elements. Normal approximations are naturally connected with central limit theorems (CLTs), i.e. convergence results displaying a Gaussian limit, and are one of the leading themes of the whole theory of probability.

The main thread of our text concerns the normal approximations, as well as the corresponding CLTs, associated with random variables that are functionals of a given Gaussian field, such as a (fractional) Brownian motion on the real line. In particular, a pivotal role will be played by the elements of the socalled Gaussian Wiener chaos. The concept of Wiener chaos generalizes to an infinite-dimensional setting the properties of the Hermite polynomials (which are the orthogonal polynomials associated with the one-dimensional Gaussian distribution), and is now a crucial object in several branches of theoretical and applied Gaussian analysis.

The cornerstone of our book is the combination of two probabilistic techniques, namely the Malliavin calculus of variations and Stein's method for probabilistic approximations.

The Malliavin calculus of variations is an infinite-dimensional differential calculus, whose operators act on functionals of general Gaussian processes. Initiated by Paul Malliavin (starting from the seminal paper [69], which focused on a probabilistic proof of Hörmander's ‘sum of squares’ theorem), this theory is based on a powerful use of infinite-dimensional integration by parts formulae.