A theorem of Gromov states that any contact manifold of
dimension 2n+1 inherits the contact structure from and embedding in the
sphere of dimension 6n+3 with its standard contact structure. We show
that if the manifold is closed the dimension of the sphere can be
decreased to 4n+3. Moreover, the embedding has other nice properties
giving in particular a geometric description of Giroux' contact open
book decompositions. A substantial part of the talk will be an
introduction to contact geometry.