  The first half of this chapter describes the development in mathematical
models of Brownian motion after Einstein's seminal papers and current
applications to optical tweezers. This instrument of choice among
single-molecule biophysicists is also an instrument of precision that requires
an understanding of Brownian motion beyond Einstein's. This is illustrated with
some applications, current and potential, and it is shown how addition of a
controlled forced motion on the nano-scale of the tweezed object's thermal
motion can improve the calibration of the instrument in general, and make it
possible also in complex surroundings. The second half of the present chapter,
starting with Sect. 9, describes the co-evolution of biological motility models
with models of Brownian motion, including very recent results for how to derive
cell-type-specific motility models from experimental cell trajectories.
