Let Y be a smooth curve embedded in a complex projective manifold X of dimension n≥2 with ample normal bundle NY|X . For every p≥0 let αp denote the natural restriction maps Pic(X)→Pic(Y(p)), where Y(p) is the p-th infinitesimal neighbourhood of Y in X. First one proves that for every p≥1 there is an isomorphism of abelian groups Coker ( α p ) ≅ Coker ( α 0 ) ⊕ K p ( Y , X ) , where Kp(Y,X) is a quotient of the C-vector space L p ( Y , X ) : = ⨁ i = 1 p H 1 ( Y , S i ( N Y | X ) * ) by a free subgroup of L p ( Y , X ) of rank strictly less than the Picard number of X. Then one shows that L 1 ( Y , X ) = 0 if and only if Y≅P1 and N Y | X ≅ 𝒪 P 1 ( 1 ) ⊕ n - 1 (i.e. Y is a quasi-line in the terminology of [4]). The special curves in question are by definition those for which dim C L1(Y,X)=1 . This equality is closely related with a beautiful classical result of B. Segre [25]. It turns out that Y is special if and only if either Y≅P1 and NY|X≅𝒪P 1(2)⊕ 𝒪 P 1 ( 1 ) ⊕ n - 2 , or Y is elliptic and deg(NY|X)=1 . After proving some general results on manifolds of dimension n ≥ 2 carrying special rational curves (e.g. they form a subclass of the class of rationally connected manifolds which is stable under small projective deformations), a complete birational classification of pairs ( X , Y ) with X surface and Y special is given. Finally, one gives several examples of special rational curves in dimension n ≥ 3.