Experimental verification of the theoretical predictions made by Albert
Einstein in his paper, published in 1905, on the molecular mechanisms of
Brownian motion established the existence of atoms. In the last 100 years
discoveries of many facets of the ubiquitous Brownian motion has revolutionized
our fundamental understanding of the role of {\it thermal fluctuations} in the
exotic structures and complex dynamics exhibited by soft matter like, for
example, colloids, gels, etc. The domain of Brownian motion transcends the
traditional disciplinary boundaries of physics and has become an area of
multi-disciplinary research. Brownian motion finds applications also in earth
and environmental sciences as well as life sciences. Nature exploits Brownian
motion for running many dynamical processes that are crucial for sustaining
life. In the first one-third of this article I present a brief historical
survey of the initial period, including works of Brown and Einstein. In the
next one-third I introduce the main concepts and the essential theoretical
techniques used for studying translational as well as rotational Brownian
motions and the effects of time-independent potentials. In the last one-third
of this article I discuss some contemporary problems on Brownian motion in
time-dependent potentials, namely, {\it stochastic resonance} and {\it Brownian
ratchet}, two of the hottest topics in this area of interdisciplinary research.Comment: 15 pages, LATEX, Based on the inaugural lecture in the Horizon
Lecture Series organized by the Physics Society of I.I.T. Kanpur, in the
"World Year of Physics 2005
