We consider a sequence of independent random variables {X n ; n = 1, 2,…). We shall suppose for simplicity that these variables have a common distribution with zero mean and nonzero variance σ2 < ∞. If \({S_n} = \sum\limits_{j = 1}^n {{X_j}}\) and \({F_n}\left( x \right) = P\left( {{S_n} < x\sigma \sqrt n } \right)\), the assumptions imply that F n (x) → Φ(x). If X1 has a lattice distribution it is natural to complement this assertion with its local analogues and to consider the asymptotic behavior of the probability P(S n = na + Nh) when X1 has a lattice distribution with the possible values a + Nh (N = 0, ± 1, ± 2,…) or the asymptotic behavior of the density p n (x) of the random variable \({S_\pi }/\sigma \sqrt n \) when X1 has an absolutely continuous distribution. We begin with local limit theorems for lattice distributions. The simplest example is the classical de Moivre-Laplace theorem, which may be interpreted as a local limit theorem for sums of independent random variables having a common distribution with two values.