  The survey is dedicated to a celebrated series of quantitave results,
developed by the Lithuanian school of probability, on the normal approximation
for a real-valued random variable. The key ingredient is a bound on cumulants
of the type $|\kappa_j(X)| \leq j!^{1+\gamma} /\Delta^{j-2}$, which is weaker
than Cram\'er's condition of finite exponential moments. We give a
self-contained proof of some of the "main lemmas" in a book by Saulis and
Statulevi\v{c}ius (1989), and an accessible introduction to the Cram\'er-Petrov
series. In addition, we explain relations with heavy-tailed Weibull variables,
moderate deviations, and mod-phi convergence. We discuss some methods for
bounding cumulants such as summability of mixed cumulants and dependency
graphs, and briefly review a few recent applications of the method of cumulants
for the normal approximation.
