In this PhD dissertation we discuss, develop and apply this fascinating theory connected to modern harmonic analysis. In particular, we make new estimations of Vilenkin-Fourier Dirichlet kernel, Cesaro kernel and prove some new results concerning boundedness of maximal operators of Cesaro means ` in the variable parameter setting. Moreover, we prove the almost everywhere convergence of Cesaro means with respect to the Vilenkin system in the vari- able parameter setting. Besides, we also do a similar investigation for the generalized Marcinkiewicz means with respect to Vilenkin-like systems.