Let $x_1,x_2,\dots$ be a sequence of independent identically distributed random variables with zero means and unit variances. Put $$ F_n(x)=\mathbf P\{(x_1+\dots+x_n)/\sqrt n<x\}. $$ Conditions are given which are necessary and sufficient for the relation $$ F_n(x)=\sum_{\nu=0}^{s-2}n^{-\nu/2}f_\nu(x)+O(\varepsilon_n),\quad n\to\infty, $$ to hold uniformly in $x$, where $s\ge2$, the sequence $\varepsilon_n$ is such that $$ \varepsilon_nn^{(s-2)/2}\to0,\quad\varepsilon_n\ge n^{-(s-1)/2},\quad n\to\infty, $$ the functions $t_\nu(x)$ are independent of $n$ and satisfy some conditions at the origin. We consider also local limit theorems.