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.