Аннотация:It is shown that 2π periodic functions whose (r-1)-th derivatives have bounded variation (r > 0) can be approximated by de La Vallée-Poussin sums σ n,m (an ⩽m =m (n) ⩽An,0 <a<A<1) at almost all points with a rate o(n−r). For functions belonging to the class Lip (α, L) (0 <α < 1), any natural N, and a positive ɛ, we have almost everywhere |f(x)−σn,m(f;x)|⩽c(f,x)n−αlnn…ln1+εNn, where lnkx=ln…lnxk(k=1,2,…) . For any triangular method of summation T with bounded coefficients we construct functions belonging to Lip (α, L) (0 < α < 1) and such that almost everywhere, lim−−−n→∞|f(x)−τn(f;x)|na(lnn…lnNn)−a=∞ where the τn(f; x) are the means of the method T.