  In the first part of this paper we describe the status of the art of this
subject. In the second part we present and motivate some new results. Indeed,
we introduce some new weighted maximal operators of the partial sums of the
Walsh-Fourier series. We prove that for some "optimal" weights these new
operators indeed are bounded from the martingale Hardy space $H_{1}(G)$ to the
space $\text{weak}-L_{1}(G),$ but is not bounded from $H_{1}(G)$ to the space
$L_{1}(G).$
