Describing bedload transport as a stochastic process is an idea that emerged in the 1930s with the pioneering work of Einstein. For a long time, the stochastic approach attracted marginal attention, but the situation has radically changed over the last decade with the recent advances in the theory of bedload transport and high-resolution measurements of sediment transport rates. This paper outlines the main achievements in our approach to bedload transport in mountain streams under low to moderate flow conditions.