We study hyper-Kahler cones and their corresponding quaternion-Kahler space
s. We present a classification of 4(n - 1)-dimensional quaternion-Kahler sp
aces with n abelian quaternionic isometries, based on dualizing superconfor
mal tensor multiplets. These manifolds characterize the geometry of the hyp
ermultiplet sector of perturbative moduli spaces of type-II strings compact
ified on a Calabi-Yau manifold. As an example of our construction, we study
the universal hypermultiplet in detail, and give three inequivalent tensor
multiplet descriptions. We also comment on the construction of quaternion-
Kahler manifolds that may describe instanton corrections to the moduli spac
e.