  We formalize the concepts of holomorphic affine and projective structures
along the leaves of holomorphic foliations by curves on complex manifolds. We
show that many foliations admit such structures, we provide local normal forms
for them at singular points of the foliation, and we prove some index formulae
in the case where the ambient manifold is compact. As a consequence of these,
we establish that a regular foliation of general type on a compact algebraic
manifold of even dimension does not admit a foliated projective structure.
Finally, we classify foliated affine and projective structures along regular
foliations on compact complex surfaces.
