By a modification of the method that was applied in study of Korolev & Shevtsova (2009), here the inequalities and are proved for the uniform distance ρ(F _n ,Φ) between the standard normal distribution function Φ and the distribution function F _n of the normalized sum of an arbitrary number n≥1 of independent identically distributed random variables with zero mean, unit variance, and finite third absolute moment β³. The first of these two inequalities is a structural improvement of the classical Berry–Esseen inequality and as well sharpens the best known upper estimate of the absolute constant in the classical Berry–Esseen inequality since 0.33477(β³+0.429)≤0.33477(1+0.429)β³<0.4784β³ by virtue of the condition β³≥1. The latter inequality is applied to lowering the upper estimate of the absolute constant in the analog of the Berry–Esseen inequality for Poisson random sums to 0.3041 which is strictly less than the least possible value 0.4097… of the absolute constant in the classical Berry–Esseen inequality. As corollaries, the estimates of the rate of convergence in limit theorems for compound mixed Poisson distributions are refined.

This paper describes the history of the search for unconditional and conditional upper bounds of the absolute constant in the Berry–Esseen inequality for sums of independent identically distributed random variables. Computational procedures are described. New estimates are presented from which it follows that the absolute constant in the classical Berry–Esseen inequality does not exceed 0.5129.

The upper bound of the absolute constant in the classical Berry–Esseen inequality for sums of independent identically distributed random variables with finite third moments is lowered to $C\leqslant 0.7056$.

A detailed classification of the asymptotic constants in the Berry–Esseen–Katz inequality for the sums of independent identically distributed random variables with finite absolute moments of order $2+\delta$ with $\delta\in[0,1]$ is proposed. Two-sided bounds and/or exact values for the asymptotically best and asymptotically exact constants are given. The lower bounds for the exact constants for the case $\delta\in[0,1)$ are presented for the first time. Also, two-sided bounds for symmetric distributions are obtained.