Normal approximations with Malliavin calculus : from Steins method to universality Steins method is a collection of probabilistic techniques that allow one to assess the distance between two probability distributions by means of differential operators. In 2007 the authors discovered that one can combine Steins method with the powerful Malliavin calculus of variations in order to deduce quantitative central limit theorems involving functionals of general Gaussian fields. This book provides an ideal introduction both to Steins method and Malliavin calculus from the standpoint of normal approximations on a Gaussian space. Many recent developments and applications are studied in detail for instance: fourth moment theorems on the Wiener chaos density estimates Breuer Major theorems for fractional processes recursive cumulant computations optimal rates and universality results for homogeneous sums. Largely self contained the book is perfect for self study. It will appeal to researchers and graduate students in probability and statistics especially those who wish to understand the connections between Steins method and Malliavin calculus 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