Those large granular lymphocytes seen in blood smears that look quite different from most "small lymphocytes" are likely natural killer or NK cells. It was initially thought that they were non-specific lymphocytes of the innate system that kill intracellularly infected cells and altered cells (i.e., tumor cells). Still, it is now known that they have many more functions. Besides their cytolytic functions, it is now known that they have multiple receptors and play a role in self-tolerance. They can also evolve memory cells, which could place them in adaptive immunity. The image depicts some of the functions of NK cells.
2. Vivier, E., et al. (2011). “Innate or adaptive immunity? The example of natural killer cells.” Science, 331: 44-9. doi:10.1126/science.1198687