  We give a negative answer to a question posed by Severi in 1951, whether the
Abelian Varieties are the only projective manifolds with trivial Chern classes.
  By Yau' s celebrated result, compact K\"ahler manifolds with trivial Chern
classes must be flat, that is, they belong to the class of Hyperelliptic
Manifolds (quotients $T/G$ of a complex torus $T$ by the free action of a
finite group $G$).
  We exhibit simple examples of projective Hyperelliptic Manifolds which are
not Abelian varieties and whose Chern classes are zero not only in integral
cohomology, but also in the Chow ring.
  We prove moreover that the Bagnera-de Franchis manifolds (quotients $T/G$ as
above but where the group $ G$ is cyclic) have topologically trivial tangent
bundle.
  Our results naturally lead to the question of classifying all compact
K\"ahler manifolds with topologically trivial tangent bundle, and all the
counterexamples to Severi's question.
