 Given a concept of distance on a manifold it is possible to talk about geodesics—a geodesic being a curve of shortest length between two points. Existence and uniqueness questions can be raised, and often answered. A significant advance was made independently by Tullio Levi-Civita in 1917 and Hermann Weyl in 1918, inspired by Einstein’s theory of general relativity, when they showed how to define parallelism on a curved manifold (on Levi-Civita’s contribution, see Bottazzini 1999 and on Weyl’s contribution see Scholz 2001). Roughly speaking, in Weyl’s presentation (1918), two vectors at different points are parallel if they belong to a family of vectors along a curve that do not vary along the curve. This became referred to as a way of establishing a connection between different points, and the theory was called the theory of connections on manifolds. One can ask if a family of tangent vectors to a curve is composed of vectors parallel to the tangent vector at the starting point: if so, the curve is a natural candidate to be considered the straightest curve between its end points, because the tangent vector never accelerates along the curve. In modern differential geometry, geodesics are defined via connections.