  We study hyperkahler cones and their corresponding quaternion-Kahler spaces.
We present a classification of 4(n-1)-dimensional quaternion-Kahler spaces with
n abelian quaternionic isometries, based on dualizing superconformal tensor
multiplets. These manifolds characterize the geometry of the hypermultiplet
sector of perturbative moduli spaces of type-II strings compactified 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 space.
