  The {\it profile} of a relational structure $R$ is the function $\phi_R$
which counts for every integer $n$ the number of its $n$-element substructures
up to an isomorphism. Many counting functions are profiles. Interesting
examples come from permutation groups. Some salient facts about the behavior of
the profile are presented. Techniques from ordered sets and combinatorics
(notably the notion of well-quasi-order, the related notions of ordered
algebras, Ramsey theorem) are illustrated. Ongoing resarch suggests to view the
profile of a relational structure $R$ as the Hilbert function of some graded
algebra associated with $R$. A hint at the solution of a conjecture of
P.J.Cameron on the integrity of the ring of the orbit algebra is given. Recent
progress made with Y.Boudabbous and N.Thi\'ery on the conjecture that the
profile is a quasi-polynomial if its growth is polynomial (and the structure
has a finite kernel) are presented.
