We prove that the maximal operator of the (C, α n)-means of the one dimensional Vilenkin-Fourier series is of weak type(L 1 , L 1). Moreover, we prove the almost everywhere convergence of the (C, α n) means of integrable functions (i.e. σ αn n f −→ f), where n ∈ N α, q and n −→ ∞ for f ∈ L 1 (G m), G m is a bounded Vilenkin group, for every sequence α = (α n), 0 < α n < 1