The first half of this review describes the development in mathematical models of Brownian motion after Einstein's and Smoluchowski's seminal papers and current applications to optical tweezers. This instrument of choice among single-molecule biophysicists is also an instrument of such precision that it requires an understanding of Brownian motion beyond Einstein's and Smoluchowski's for its calibration, and can measure effects not present in their theories. This is illustrated with some applications, current and potential. It is also shown how addition of a controlled forced motion on the nano-scale of the thermal motion of the tweezed object can improve the calibration of the instrument in general, and make calibration possible also in complex surroundings. The second half of the present re-view, starting with Sect. 9, describes the co-evolution of biological motility models with models of Brownian motion, including recent results for how to derive cell-type-specific motility models from experimental cell trajectories.