Brownian motion is the most studied of all stochastic processes; it is also the basis for stochastic analysis developed in the second half of the 20th century. The fine properties of the sample path of a Brownian motion have been carefully studied, starting with the fundamental work of Paul Lévy who also considered more general processes with independent increments and extended the Brownian motion results to this class. Lévy showed that a Brownian path in d (d ≥ 2) dimensions had zero Lebesgue measure; he asked for the right Hausdorff measure function to measure the sample path. This is the starting point for my joint work with Ciesielski  in 1961 which we will summarize in this lecture. We further describe some of the papers published in the last 40 years which built on the results and methods of , focusing only on those papers which find properties of the sample path of Brownian motion.