  The aim of my thesis is to discuss, develop and apply the newest developments
of this fascinating theory connected to modern harmonic analysis. In
particular, we investigate some strong convergence result of partial sums of
Vilenkin-Fourier series. Moreover, we derive necessary and sufficient
conditions for the modulus of continuity so that norm convergence of
subsequences of Fej\'er means is valid. Furthermore, we consider Riesz and
N\"orlund logarithmic means. It is also proved that these results are the best
possible in a special sense. As applications both some well-known and new
results are pointed out. In addition, we investigate some $T$ means, which are
"inverse" summability methods of N\"orlund, but only in the case when their
coefficients are monotone.
