Continuum Mechanics (CM) is a natural field of application of concepts and methods of Differential Geometry (DG). The very foundations of both disciplines are intertwined in a deep manner. A presentation of basic issues in CM adopting the powerful tools of modern DG is still substantially lacking. This booklet is intended to contribute to fill this gap, with specific reference to Elasticity theory. The classical subject is thoroughly revisited and revised in its basic aspects and in the general context of finite deformations. A case study of rubber-like materials enlightens the new concepts introduced by the geometric theory and opens the way for applications to soft materials such as the ones of interest in biomechanics.