  We describe in detail the history of Brownian motion, as well as the
contributions of Einstein, Sutherland, Smoluchowski, Bachelier, Perrin and
Langevin to its theory. The always topical importance in physics of the theory
of Brownian motion is illustrated by recent biophysical experiments, where it
serves, for instance, for the measurement of the pulling force on a single DNA
molecule.
  In a second part, we stress the mathematical importance of the theory of
Brownian motion, illustrated by two chosen examples. The by-now classic
representation of the Newtonian potential by Brownian motion is explained in an
elementary way. We conclude with the description of recent progress seen in the
geometry of the planar Brownian curve. At its heart lie the concepts of
conformal invariance and multifractality, associated with the potential theory
of the Brownian curve itself.
