The authors investigate the approximation properties of the Feller operator \(L_n\) [W. Feller, An introduction to probability theory and its applications, Vol. II (1966; Zbl 0138.10207)], defined by \[ (L_n f)(x)= E[S_{n,x}/n]= \int^\infty_{-\infty} f\bigl({\textstyle{t\over n}}\bigr) d_t F_{n,x}(t), \] where \(F_{n,x}\) is the distribution function of the sum \(S_{n,x}= X_{1,x}+\cdots+ X_{n,x}\), \((X_{k,n})\) representing a sequence of independent and identically distributed random variables, with \(EX_{k,x}=x\) and finite variance. It is assumed that \(f\) is a real-valued and integrable function, in the Lebesgue-Stieltjes sense, with respect to \(F_{n,x}\) on \(R\). We mention that the first approximation properties of this operator of Feller were investigated in our memoir published by the reviewer [Rev. Roum. Math. Pur. Appl. 14, 673-691 (1969; Zbl 0187.32502)]. The authors establish first some inequalities involving the ordinary or weighted moduli of continuity and then they give some estimates of the order of approximation of \(f\) by \(L_nf\), using Hölder type norms.