  We present and discuss the many results obtained concerning a famous limit
theorem, the local limit theorem, which has many interfaces, with Number Theory
notably, and for which, in spite of considerable efforts, the question
concerning conditions of validity of the local limit theorem, has up to now no
satisfactory solution. These results mostly concern sufficient conditions for
the validity of the local limit theorem and its interesting variant forms:
strong local limit theorem, strong local limit theorem with convergence in
variation. Quite importantly are necessary conditions, and the results obtained
are sparse, essentially: Rozanov's necessary condition, Gamkrelidze's necessary
condition, and Mukhin's necessary and sufficient condition. Extremely useful
and instructive are the counter-examples due to Azlarov and Gamkrelidze, as
well as necessary and sufficient conditions obtained for a class of random
variables, such as Mitalauskas' characterization of the local limit theorem in
the strong form for random variables having stable limit distributions. The
method of characteristic functions and the Bernoulli part extraction method,
are presented and compared. A second part of the survey is devoted to the more
recent study of the almost sure local limit theorem, instilled by Denker and
Koch. The almost sure local limit theorems established already cover the i.i.d.
case, the stable case, Markov chains, the model of the Dickman function. Our
aim in writing this monograph was notably to bring to knowledge many
interesting results obtained by the Lithuanian and Russian Schools of
Probability during the sixties and after, and which are essentially written in
Russian, and moreover often published in Journals of difficult access.
