Using optical tweezers combined with image analysis we investigate motility of single proteins in membranes and of organelles inside living cellular organisms, one key issue being that the organisms are kept alive and healthy. Studies of two different biological systems will be presented: By specifically attaching a bead to a single protein, the lambda-receptor, which is a porin in the outer membrane of E. coli bacteria, we revealed its nanoscale diffusional motion and proposed a model that allows for extraction of the characteristic physical parameters including the diffusion constant. Surprisingly, the observed mobility is caused not only by thermal motion but in addition by an active motion associated with the metabolism of the organism. Connected to this, we show that antibiotics and antimicrobial peptides have a pronounced effect on single protein motility. The second biological system presented will be an S. pombe yeast cell, where the diffusion patterns of naturally occurring lipid granules have been uncovered using optical trapping and single particle tracking; the granules perform anomalous diffusion, with subdiffusion being most predominant at short time-lags, and the biological functions giving motility footprints at longer time-lags. The diffusional properties inside living yeast cells change during the cell cycle, and a novel maximal excursion method shows that the physical origin of the observed motility is probably fractional Brownian motion.