  Projective superspace provides a natural framework for the construction of
actions coupling hypermultiplets to conformal supergravity. We review how the
off-shell actions are formulated in superspace and then discuss how to
eliminate the infinite number of auxiliary fields to produce an on-shell N=2
supersymmetric sigma model, with the target space corresponding to a generic
4n-dimensional hyperkahler cone. We show how the component action coupling the
hypermultiplets to conformal supergravity may be constructed starting from
curved superspace. The superspace origin of the geometric data -- the
hyperkahler potential, complex structures, and any gauged isometries -- is also
addressed.
