Explore the concept of subgroup distortion in hyperbolic groups through this informative lecture. Delve into the distortion function of subgroups and its significance in measuring the difference between intrinsic word metric and induced metric from the ambient group. Examine Ol'shanskii's findings on the lack of restrictions for distortion functions in finitely presented groups. Investigate the specific case of hyperbolic groups and their subgroup distortion functions. Survey known examples of distortion functions in hyperbolic groups and discover new examples presented from joint work with Tim Riley. Gain insights into this advanced topic in geometric group theory and its implications for understanding subgroup structures within hyperbolic groups.