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Unlike the classical theory of Fourier series which deals with decompositionof a function into sinusoidal waves the Vilenkin (Walsh) functions arerectangular waves. The development of the theory of Vilenkin-Fourier series hasbeen strongly influenced by the classical theory of trigonometric series butthere are a lot of differences also. The aim of my master thesis is to discuss,develop and apply the newest developments of this fascinating theory connectedto modern harmonic analysis. In particular, we investigate N\"orlund means butonly in the case when their coefficients are monotone and prove convergence inLebesgue and Vilenkin-Lebesgue points. Since almost everywhere points areLebesgue and Vilenkin-Lebesgue points for any integrable functions we obtainalmost everywhere convergence of such summability methods.

On the almost everywhere and norm convergences of N\"orlund means with respect to Vilenkin systems