The Hopf Conjecture states that for closed, orietable, even-dimensional manifolds, the Euler characteristic is strictly positive. Results due independently to P uttmann and Searle [13] and Rong [14], and due to Rong and Su [15], showing that the Hopf Conjecture holds under the additional hypothesis of abelian symmetries. In this thesis we detail the proofs of these two results. For the rst result, we provide the details of the original proof, whereas for the second, we give a more streamlined proof that relies of the Borel formula.