Abstract: 
Around 1923, \'Elie Cartan introduced affine connections on
manifolds and defined the main related concepts: torsion, curvature,
holonomy groups. He discussed applications of these concepts in
Classical and Relativistic Mechanics; in particular he explained how
parallel transport with respect to a connection can be related to
the principle of inertia in Galilean Mechanics and, more generally,
can be used to model the motion of a particle in a gravitational
field. In subsequent papers, \'Elie Cartan extended these concepts
for other types of connections on a manifold: Euclidean, Galilean
and Minkowskian connections which can be considered as special types
of affine connections, the group of affine transformations of the
affine tangent space being replaced by a suitable subgroup; and more
generally, conformal and projective connections, associated to a
group which is no more a subgroup of the affine group.

Around 1950, Charles Ehresmann introduced connections on a fibre
bundle and, when the bundle has a Lie group as structure group,
connection forms on the associated principal bundle, with values in
the Lie algebra of the structure group. He called {\it Cartan
connections\/} the various types of connections on a manifold
previously introduced by \'E. Cartan, and explained how they can be
considered as special cases of connections on a fibre bundle with a
Lie group $G$ as structure group: the standard fibre of the bundle
is then an homogeneous space $G/G'$; its dimension is equal to that
of the base manifold; a Cartan connection determines an isomorphism
of the vector bundle tangent to the the base manifold onto the
vector bundle of vertical vectors tangent to the fibres of the
bundle along a global section.

These works are reviewed and some applications of the theory of
connections are sketched.

1. Institut de Mathématiques
   Université Pierre et Marie Curie
   4, place Jussieu
   75252 Paris Cedex 05, France