The playground for anomalous diffusion is in the classical world of physics, chemistry, biology, economics, and even art. However, the question of quantum Brownian motion (QMB) and quantum diffusion fascinated quantum scientist for decades. After a general introduction to the Brownian motion subject, I will describe some initial attempts to describe QMB and observe its quantum nature in experiments. Recently we witness incredible developments of laboratory techniques to control and manipulate quantum systems at very low temperatures, at the single particle, as well as at the many body level. This calls for and opens new path for theoretical and experimental studies of quantum anomalous diffusion, quantum non–Markovian dynamics, or quantum breakdown of ergodicity. I will describe some of these phenomena, and comment on their possible impact on quantum technologies.