Sophus Lie had a tremendous impact in several areas of mathematics. This is his monumental treatise on the theory of contact transformations. It was through the study of these transformations, which he began in his thesis, that eventually led Lie to his important work on continuous groups and on solvability of differential equations. Klein described Lie's work on contact transformations as one of the most elegant contributions to differential geometry made in recent times”. This statement holds true even today. In Lie's time, his methods were used to show how ostensibly unrelated problems in geometry (or the geometric study of differential equations) are really equivalent.