Screw theory has become a well developed tool for analysis and synthesis in mechanisms and robotics, though its development and intrinsic relationship with the mechanism development has not yet been well revealed. The relationship is to be presented, in the following through the historical development of classical mechanisms by their mathematician inventors. It reveals that a study of geometry led to development of classical mechanisms and new mechanisms while these mechanisms are conversely related to development of the mathematical foundation in the form of screw theory and screw algebra. This traces back to the 17th and 18th century and to the thriving 19th century when the study of screw theory reached its height. A comprehensive account is then presented of the latter development in the 20th century with the latest progress. The relationship between screw theory and Lie theory is further revealed via the development of finite displacement screws in the 20th century and in their development in the 1990s. This reveals the intrinsic relationship between finite displacement screws and Lie groups and leads to a relationship diagram between two theories as presented in the book “Geometrical Foundation and Screw Algebra of Mechanisms and Robotics”. The revelation leads to the review of use of screw theory for the study of stiffness and compliance of mechanisms and robotics.