I. Probability Distributions and Characteristic Functions -- ยง 1. Random variables and probability distributions -- ยง 2. Characteristic functions -- ยง 3. Inversion formulae -- ยง 4. The convergence of sequences of distributions and characteristic functions -- ยง 5. Supplement -- II. Infinitely Divisible Distributions -- ยง 1. Definition and elementary properties of infinitely divisible distributions -- ยง 2. Canonical representation of infinitely divisible characteristic functions -- ยง 3. An auxiliary theorem -- ยง 4. Supplement -- III. Some Inequalities for the Distribution of Sums of Independent Random Variables -- ยง 1. Concentration functions -- ยง 2. Inequalities for the concentration functions of sums of independent random variables -- ยง 3. Inequalities for the distribution of the maximum of sums of independent random variables -- ยง 4. Exponential estimates for the distributions of sums of independent random variables -- ยง 5. Supplement -- IV. Theorems on Convergence to Infinitely Divisible Distributions -- ยง 1. Infinitely divisible distributions as limits of the distributions of sums of independent random variables -- ยง 2. Conditions for convergence to a given infinitely divisible distribution -- ยง 3. Limit distributions of class L and stable distributions -- ยง 4. The central limit theorem -- ยง 5. Supplement -- V. Estimates of the Distance Between the Distribution of a Sum of Independent Random Variables and the Normal Distribution -- ยง 1. Estimating the nearness of functions of bounded variation by the nearness of their Fourier-Stieltjes transforms -- ยง 2. The Esseen and Berry-Esseen inequalities -- ยง 3. Generalizations of Esseenโ{128}{153}s inequality -- ยง 4. Non-uniform estimates -- ยง 5. Supplement -- VI. Asymptotic Expansions in the Central Limit Theorem -- ยง 1. Formal construction of the expansions -- ยง 2 Auxiliary propositions -- ยง 3. Asymptotic expansions of the distribution function of a sum of independent identically distributed random variables -- ยง 4. Asymptotic expansions of the distribution function of a sum of independent non-identically distributed random variables, and of the derivatives of this function -- ยง 5. Supplement -- VII. Local Limit Theorems -- ยง 1. Local limit theorems for lattice distributions -- ยง 2. Local limit theorems for densities -- ยง 3. Asymptotic expansions in local limit theorems -- ยง 4. Supplement -- VIII. Probabilities of Large Deviations -- ยง 1. Introduction -- ยง 2. Asymptotic relations connected with Cramรฉrโ{128}{153}s series -- ยง 3. Necessary and sufficient conditions for normal convergence in power zones -- ยง 4. Supplement -- IX. Laws of Large Numbers -- ยง 1. The weak law of large numbers -- ยง 2. Convergence of series of independent random variables -- ยง 3. The strong law of large numbers -- ยง 4. Convergence rates in the laws of large numbers -- ยง 5. Supplement -- X. The Law of the Iterated Logarithm -- ยง 1. Kolmogorovโ{128}{153}s theorem -- ยง 2. Generalization of Kolmogorovโ{128}{153}s theorem -- ยง 3. The central limit theorem and the law of the iterated logarithm -- ยง 4. Supplement -- Notes on Sources in the Literature -- References -- Subject Indes -- Table of Symbols and Abbreviations