The classical local limit theorem in mathematics is Abraham de Moivre's theorem for binomial probabilities in 1733, which was published in the second edition of his Doctrine of Chance. His theorem is taught in elementary classes on probability and serves as a starting point for our discussion. The theory has been developed for independent random variables mainly by the russian school around Kolmogorov using Fourier analysis and for Markov chains by Nagaev using spectral analysis. New developments include time series arising in dynamics. I am planning to discuss in addition two applications, one to conservativity problems for group extensions and the other one to the rate of convergence of Poincare series in hyperbolic geometry. If time permits I will also present a recent new outlook on de Moivre's theorem.