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Given integers r>1, n>1 and q> n-2, we consider projective varieties X ofdimension r+1 such that through n generic points of X passes a rational curveof degree q, contained in X. More precisely, we study the class X_{r+1,n}(q) ofsuch varieties which moreover generate a projective space of the maximaldimension.  We determine all varieties of a class X_{r+1,n}(q) when q is not equal to2n-3. In particuliar, we show that there exists a variety X' in P^{r+n-1}, ofminimal degree and a birational map F: X'---> X which sends a generic sectionof X' by a P^{n-1} onto a rational normal curve of degree q.  Without hypothesis on q, we define a quasi-grassmannian structure on thespace of the rational normal curves of degree q contained in a variety X of theclass X_{r+1,n}(q). We prove that X is of the form described above if and onlyif this quasi-grassmannian structure is flat.  We also give examples of varieties of the classes X_{r+1,3}(3) etX_{r+1,4}(5) which are not of this form.

Sur les vari\'et\'es X dans P^N telles que par n points passe une courbe de X de degr\'e donn\'e