  We present theories of N=2 hypermultiplets in four spacetime dimensions that
are invariant under rigid or local superconformal symmetries. The target spaces
of theories with rigid superconformal invariance are (4n)-dimensional {\it
special} hyper-K\"ahler manifolds. Such manifolds can be described as cones
over tri-Sasakian metrics and are locally the product of a flat
four-dimensional space and a quaternionic manifold. The latter manifolds appear
in the coupling of hypermultiplets to N=2 supergravity. We employ local
sections of an Sp$(n)\times{\rm Sp}(1)$ bundle in the formulation of the
Lagrangian and transformation rules, thus allowing for arbitrary
coordinatizations of the hyper-K\"ahler and quaternionic manifolds.
