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Theory of relations is the framework of this thesis. It is about enumerationof finite structures. Let $\mathscr C$ be a class of finite combinatorialstructures, the \emph{profile} of $\mathscr C$ is the function$\varphi_{\mathscr C}$ which count, for every $n$, the number of members of$\mathscr{C}$ defined on $n$ elements, isomorphic structures been identified.The generating function for $\mathscr C$ is $\mathcal H_{\mathscrC}(X):=\sum_{n\geqq 0}\varphi_{\mathscr C}(n)X^n$. Many results about thebehavior of the function $\varphi_{\mathscr C}$ have been obtained. Albert andAtkinson have shown that the generating series of the profile of some classesof permutations are algebraic. we show how this result extends to classes ofordered binary structures using the notions of theory of relations. This is thesubject of the first part of this thesis. The second part is concerned with thenotion of minimality. An hereditary class of finite structures is minimal if itis infinite and every proper hereditary subclass is finite. We show, inparticular, that ind-minimal classes are wqo ages and their number is thecontinuum.  The last part is motivated by the surprising phenomenon of the \emph{jump}observed in the behavior of the profile of hereditary classes of finitestructures. We show that the profile of an hereditary classe made of orderedstructures which have finite monomorphic decomposition is a polynomial. We alsoshow that if the profile of a hereditary class of finite ordered binarystructures is not bounded by a polynomial then it is at least exponential. Thisresult generalizes the result obtained by Balogh, Bollob\'as and Morris (2006)for ordered graphs.

Sur l'\'enum\'eration de structures discr\`etes, une approche par la th\'eorie des relations