In this paper we discuss an analogy of the Carleson-Hunt theorem with respect to Vilenkin systems. In particular, we investigate the almost everywhere convergence of Vilenkin-Fourier series of f∈Lp(Gm) for p>1 in case the Vilenkin system is bounded. Moreover, we state an analogy of the Kolmogorov theorem for p=1 and construct a function f∈L1(Gm) such that the partial sums with respect to Vilenkin systems diverge everywhere.