The author gives detailed derivations of the diffusion coefficient in various physical settings: in Einstein’s original setting, for particles subjected to an external force, for non-constant temperatures, for both external force and non-constant temperature, and for interacting particles. The method differs from Einstein’s original approach insofar as it does not use a fictitious force but concentrates on the dynamic equilibrium between the forces of pressure and friction acting upon a Brownian particle. One advantage of this approach is that the argument can easily be generalized to more difficult physical settings and leads to alternative derivations of and new insights into known principles and effects in mathematical physics (e.g., the Smoluchowski approximation, the Ludwig-Soret and Enskog-Chapman effects etc.). The style of the paper is expository, containing a good bibliography and many historical remarks.