  Stein's method is a method of probability approximation which hinges on the
solution of a functional equation. For normal approximation the functional
equation is a first order differential equation. Malliavin calculus is an
infinite-dimensional differential calculus whose operators act on functionals
of general Gaussian processes. Nourdin and Peccati (2009) established a
fundamental connection between Stein's method for normal approximation and
Malliavin calculus through integration by parts. This connection is exploited
to obtain error bounds in total variation in central limit theorems for
functionals of general Gaussian processes. Of particular interest is the fourth
moment theorem which provides error bounds of the order
$\sqrt{\mathbb{E}(F_n^4)-3}$ in the central limit theorem for elements
$\{F_n\}_{n\ge 1}$ of Wiener chaos of any fixed order such that
$\mathbb{E}(F_n^2) = 1$. This paper is an exposition of the work of Nourdin and
Peccati with a brief introduction to Stein's method and Malliavin calculus. It
is based on a lecture delivered at the Annual Meeting of the Vietnam Institute
for Advanced Study in Mathematics in July 2014.
