In 1905, Einstein derived in his first paper on Brownian motion the famous formula for the diffusion coefficient of a particle suspended in a liquid, which involves Avogadro’s number. While Poincaré in 1904 still believed that Brownian motion contradicted Carnot’s principle, Einstein’s aim was to establish the existence of atoms and to vindicate the kinetic theory of heat. Experimental confirmation came in 1908 from Perrin, meanwhile Langevin invented stochastic calculus. This led Wiener in 1923 to introduce his measure on continuous random paths, providing the first rigorous mathematical theory of Brownian motion (with Bachelier in 1900!), as well as the first example of a functional integral. Brownian paths are everywhere nowadays, as a fundamental common thread linking mathematics and physics. They form the geometrical and probabilistic texture of quantum mechanics and field theory, while Brownian fluctuations are also used in biology to measure forces with high precision. Recently, significant advances in the random geometry of the planar Brownian curve have occurred in mathematics, through the introduction of the Stochastic Loewner Evolution, and in parallel in physics through the use of conformal field theory and quantum gravity. I will attempt to describe some of these various aspects, and the importance of Brownian motion for the development of the Natural and Mathematical Sciences.