Let M be a smooth projective curve of genus g > 0. Let J be the Jacobian of M and let M ) be the i th symmetric product. Fix a base point P e M and define a map ~bi: M ) ~ J by Oi(D) = D iP. Mattuck [6, 7] has shown that, if i > 2g 1, then 0i is a Ipi-g-bundle. The most interesting case occurs when i = 2g 1, since in [1 ] the bundles for larger i are determined by this one, The inversion of abelian integrals problem asks: What is an explicit description of the transition functions of the bundle 0~ ? For an excellent introduction to the problem, see Kempf 's article [5]. Until now, the only complete answer was given in genus 1 by the Abel Inversion Theorem, the Riemann-Roch Theorem, and Riemann's approach through theta functions. Even without the transition functions, Gunning [1, 2] and Kempf [3,4] were able to extract a great deal of information about M from the bundles 0,. In this paper, we present a solution to the inversion of abelian integrals problem in the cases (i) curves of genus 2 (ii) non-hyperelliptic curves o f genus 3. The techniques we use are very geometric but essentially elementary. We strongly emphasize the role played by effective divisors and by the Riemann-Roch Theorem. To build sections, we use the commutative diagram